jagomart
digital resources
picture1_Matrix Pdf 172769 | Dgt Matrices


 162x       Filetype PDF       File size 2.12 MB       Source: dgtstudy.com


File: Matrix Pdf 172769 | Dgt Matrices
1 dgt mh cet 12th mathematics study material matrices 18 02 matrices iv square matrix amatrix in which number of syllabus rows is equal to the number of columns is ...

icon picture PDF Filetype PDF | Posted on 27 Jan 2023 | 2 years ago
Partial capture of text on file.
                                                                                                                                                      1
                                            DGT MH –CET 12th MATHEMATICS  Study Material
                                                                            Matrices                                               18
                          02                                               Matrices
                                                                                       iv. Square Matrix : Amatrix in which number of
                      Syllabus                                                             rows is equal to the number of columns, is
                                                                                           called a square matrix. The elements a  of a
                            Types of Matrices  Algebra of Matrices                                                                ij
                                                                                           square matrix A = [a ) m × m for which i = j
                            Equality of Two Matrices  Trace of a Matrix.                                      ij
                                                                                           i.e. the elements a , a ··· a   are called the
                            Equivalent Matrix  Inverse of a Matrix.                                         11  22    mm 
                            Applications of Matrices                                       diagonal elements and the line along which
                                                                                           called the principal diagonal or leading diagonal
                            In Mathematics, a matrix (plural matrices) is a                of the matrix;
                            rectangular array of numbers, symbols or                             1 2 3
                            expressions, arranged in rows and columns. The                               
                                                                                                         
                                                                                                 3 2 1
                            individualsin a matrix are called its elements or          e.g.A =               is a square matrix of order
                                                                                                2   3 1
                            entries. Generally, matrix is written in the following                       33
                            way :                                                      in which diagonal elements are 1,2, l.
                                                                                       v. Null Matrix or Zero Matrix A matrix of order
                                                                                           m × n whose all elements are zero is called a
                                     a     a ...  a                                        null matrix of order m × n.
                                      11    12     1n                                      It is denoted by 0.
                                     a21   a22... a2n                                               0 0        0 0 0
                               A =                 = [a ] m × n                        e.g. 0 =       and           
                                                           ij                                       0 0        0 0 0
                                     am1   a  amn
                                            m2                                             are two null matrices of order 2×2 and 2 ×3,
                            where, a  is the entry at ith row andjth column.               respectively.
                                     ij
                            The orderofamatrixAismx n,wheremis the number              vi. Diagonal Matrix A square matrix is called a
                            of rows and n is the number of columns.                        diagonal matrix, if all its non-diagonal elements
                            Types of Matrices                                              are zero and diagonal elements mayor may
                            i. Row Matrix  : A matrix which has only one                   not be zero.
                               row and any nuymber of columns, is called a                 If d , d , d ......,d  are elements of principal
                               row matrix.                                                     1  2   3      n
                                                                                           diagonal of a diagonal matrix of order n x n,
                               e.g. A = [27 85 1 4]     is a row matrix.                   then matrix is denoted as diag [d , d ,...... d ]
                                                    1 × 4                                                                    1  2      n
                            ii. Column Matrix Amatrix is said to be a column                    a   0 0
                               matrix, if it has only one column and any                                 
                               number of rows.                                         e.g. A = 0   b 0 is a diagonal matrix which  is
                                                                                                0   0 c
                                         1                                                             
                                         2                                               a diagonal matrix which  is denoted by A =
                               e.g. A =       is  a column matrix.                       diag [a, b, c].
                                         3
                                          31                                        Note :  The number of zeroes in a diagonal matrix
                            iii. Rectangular Matrix Amatrix in which number                lie between n2 – n to n2, where n is an order of
                               of rows is not equal to the number of columns               the matrix.
                               or vice-versa is called a rectangular matrix.           vii: Triple-Diagonal Matrix A square matrix A is
                                          1 2 3                                            said to be a triple-diagonal matrix, if all its
                               e.g. A =           is a rectangular matrix of             elements are zero except possibly for those
                                        4   5 6
                                                                                         lying on the principal diagonal, the diagonal
                               order 2 × 3.                                                immediately above as well as below the
                                                                                           principal diagonal.
                      MATHEMATICS – MHT-CET                                                        Himalaya Publication Pvt. Ltd.
         DGT Group - Tuitions (Feed Concepts)   XIth – XIIth | JEE | CET | NEET  |  Call : 9920154035 / 8169861448
                                                                                                                                                                                                                                                                                                                                                                                                2
                                                                                                                DGT MH –CET 12th MATHEMATICS  Study Material
                                                                                                                                                                                                Matrices                                                                                                                                       19
                                                                                                                                                    1             1            0           0                               matrix A (not all) is known as sub matrix of A i.e.
                                                                                                 5                       0                                                                                               the matrix B constituted by the array of elements,
                                                                                                                                                 1              2           1 0                                         which are left after deleting some rows or columns
                                                                                                  3 4 3                                                                                                                     or both of matrix A is called submatrix of A.
                                                                      e.g.  A =                                                   and   0                        1            2           3                               (a) Principal Submatrix A square submatrix B
                                                                                                 0             0          4                                                                  
                                                                                                                                                  0              0            4          5                                of a square matrix A is called a principal submatrix,
                                                                      viii. Scalar Matrix. A square matrix A = [aij Iis                                                                                                       if the diagonal elements of B are also diagonal
                                                                               said to be scalar matrix, if                                                                                                                   elements of A.
                                                                      (a) a = 0,                            0,i j                                                                                                         (b) Leading Submatrix A principal square
                                                                                  ij                   ij                                                                                                                     submatrix B is said to be a leading submatrix of a
                                                                      (b)a = 0, i  h, wherek  0                                                                                                                            square matrix A, if it is obtained by deleting only
                                                                                  ij                                                                                                                                          some of the last rows and the corresponding
                                                                      In other words, a diagonal matrix is said to be a
                                                                      scalar matrix, if the elements of principal diagonal                                                                                                    columns such that the leading elements (i.e. au) is
                                                                      are same.                                                                                                                                               not lost
                                                                                                      5 0 0                                                                                                                 xiii. Horizontal Matrix Any matrix in which the
                                                                                                                                                                                                                            number of columns is more than the number of
                                                                               e.g. A = 0                        5 0 is a scalar matrix.                                                                                    rows is called a horizontal matrix.
                                                                                                      0          0 5
                                                                                                                                                                                                                                           2           3            4            5 
                                                                      ix. Limit Matrix or Identity Matrix                                                                                                                                                                                  
                                                                                                                                                                                                                                                8 9                    7            2
                                                                      A square matrix A = [a  is said to be a unit matrix                                                                                                     e.g.                                                          is a horizontal matrix.
                                                                                                                                 ijl                                                                                                         2           2 3 4 
                                                                      or identity matrix, if                                                                                                                                                                                               
                                                                      (a) a  = 0, i  j                                                                                                                                      xiv.Vertical Matrix  Any matrix in which the
                                                                                  ij                                                                                                                                          number of rows is more than the number of
                                                                      (b) a  = 1, i  j
                                                                                   ij                                                                                                                                         columns is called column matrix.
                                                                      In other words, A diagonal matrix, whose elements                                                                                                                        2 3
                                                                      of principal diagonal are equal to 1 and all                                                                                                                                         
                                                                      remaining elements are zero, is known as unit or                                                                                                                      4          5
                                                                      identity matrix. It is denoted by 1.                                                                                                                    e.g.                                   is a column matrix.
                                                                                                                                                                                                                                            6          7
                                                                                                                                                                                                                                                           32
                                                                                                 1          0 0                                                                                                             Algebra of Matrices
                                                                                                 0          1 0                                                                                                             Four types of algebra of matrices are defined
                                                                      e.g. I =                                             is a unit matrix of order 3.                                                                     below:
                                                                                                 0          0 1
                                                                                                                                                                                                              1.            Addition of Two Matrices
                                                                      x. Upper Triangular Matrix                                                                                                                              Let A = [a ]                            and B = [b ]                              are two matrices
                                                                      A square matrix A = [a  is known as upper                                                                                                                                           ij  m×n                                   ij  m×n
                                                                      triangular matrix, if                                                ij                                                                                 whose orders are same, then
                                                                                                                                                                                                                              A + B = [a + b ] i1,2....,mand j1,2,....n
                                                                                   a  = 0,  i  j                                                                                                                                                         ij         ij
                                                                                         ij                                                                                                                                   Example 1
                                                                                              0          1 0                                                                                                                                                                                   
                                                                                                                                                                                                                                                   2 3                        5 1
                                                                      e.g. A =   0                       1 0  is an upper triangular matrix.                                                                                If A =   0                                    3             0   and
                                                                                              0          0 1                                                                                                                                                                                   
                                                                                                                      
                                                                      xi. Lower Triangular Matrix A square matrix                                                                                                                                  2                   5 1 
                                                                      A = [a ] is  known as triangular matrix, if [a   = 0                                                                                                             B = 2 3                                 1/2 then A + B is
                                                                                       ij                                                                                                ij                                                                                               
                                                                       i j
                                                                                              1          0 0                                                                                                                                                                         
                                                                                                                                                                                                                                         2 3                        5 1
                                                                                                 4 2 0                                                                                                                        a.        0                        3              0 
                                                                      e.g. A =                                          is a lower triangular matrix.                                                                                                                                
                                                                                              5          6 3
                                                                                                                                                                                                                                                                                      
                                                                      xii. Submatrices of a Matrix A matrix B obtained                                                                                                        b.  3 1 5 1 
                                                                      by deleting the row (s) or column (s) or both of a                                                                                                                2               6                     1/2
                                                        MATHEMATICS – MHT-CET                                                                                                                                                                               Himalaya Publication Pvt. Ltd.
                       DGT Group - Tuitions (Feed Concepts)   XIth – XIIth | JEE | CET | NEET  |  Call : 9920154035 / 8169861448
                                                                                                                                                                                                                                                                                                                                                                                                3
                                                                                                                DGT MH –CET 12th MATHEMATICS  Study Material
                                                                                                                                                                                                Matrices                                                                                                                                       20
                                                                                2 3 1 5 0                                                                                                                                           1 7                                                       3            1 
                                                                      c.                                                                                                                                                    c.                                                        d.                          
                                                                                   0                      6                      1/2                                                                                                       5 7                                                         5 3
                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                               2 4                           1           3
                                                                                2 3 1 5 1                                                                                                                   Sol (b) A – B =                                               –                          
                                                                                                                                                                                                                                                                                                       
                                                                      d.                                                                                                                                                                                        3 2                         2 5
                                                                                0                        6                      1/2                                                                                                                                                                    
                                                        Sol (c) Since, A and B are of the same order 2 × 3.                                                                                                                                            21                   43                         1          1
                                                                                                                                                                                                                                        =                                                 =                           
                                                                      Therefore, addition of A and B is defined and is                                                                                                                                                                                                
                                                                                                                                                                                                                                                  3(2) 25                                             5 3
                                                                      given by                                                                                                                                                                                                                                        
                                                                                                                                                                                                                3.            Scalar Multiplication
                                                                                                                                                                                                                              Let A = [a ] be any m×n matrix and k be any
                                                                                                                                                                                                                                                           ij
                                                                                              2 3 1 5 11                                                                                                                 scalar. Then, the matrix obtained by multiplying
                                                                      A + B =                                                                            1                                                                  each element of A by k is called the scalar
                                                                                              22                       33                    0                                                                           multiplication of A by k and it is denoted by kA.
                                                                                                                                                         2
                                                                                                                                                                                                                            Thus, if A = [a ]                                  , then kA = [ka ]
                                                                                                                                                                                                                                                                      ij  m×n                                             ij  m×n
                                                                                     2 3 1 5 0                                                                                                                                                              1 2 3                                                         2 4 6
                                                                                                                                                                                                                                                                                                                                                
                                                                               =                                                      1                                                                                                                     3          2 1                                             6          4 2
                                                                                     0                         6                                                                                                            e.g. If A =                                            , then 2A =                                                  
                                                                                                                                      2                                                                                                                     1          3 1                                             2          6 2
                                                                                                                                                                                                                                                                                                                                                  
                                                        Properties of Addition of Matrices                                                                                                                                    Properties of Scalar Multiplication
                                                                      Let A, Band C are three matrices of same order,                                                                                                         If A = [a ]                          and B = [b ]                                 are two matrices
                                                                      then                                                                                                                                                                            ij  m×n                                     ij  m x n
                                                        i.            Matrix addition is commutative                                                                                                                          and ,  are   two scalars, then
                                                                      i.e.A + B = B + A                                                                                                                         i.            (A + B) = A  + AB
                                                        ii.           Matrix addition is associative,                                                                                                           ii.           (A + ) A =  A + A
                                                                      i.e. (A + B) + C = A + (B + C)                                                                                                            iii.          (+ ) A = A (A) = ( A)
                                                        iii.          If 0 is a null matrix of order m × n and                                                                                                  iv.           (–) A = – (A) = A(–)
                                                                      A + 0 =A =0 + A, then 0 is known as additive                                                                                              4.            Multiplication of Two Matrices
                                                                                                                                                                                                                              If A = [a  m × n and B = [b ]                                                      are two matrices
                                                                      identity.                                                                                                                                                                     ij                                              ij  m×n
                                                        iv.           If for each matrix A = [a ]                                                       a matrix (–A) is                                                      such that the number of columns of A is equal to
                                                                      such that                                                             ij  m×n                                                                           the number of rows of B, then a matrix
                                                                                                                                                                                                                              C = [c ]                        of order m x p is known as product
                                                                               A + (– A) = 0 = (–A) + A,                                                                                                                                       ij  m × p
                                                                      then matrix (– A) is known as additive inverse of                                                                                                       of matrices A and B,  where
                                                                      A .                                                                                                                                                              c             n
                                                                                                                                                                                                                                          ij  =  a b j b a b ...a b
                                                        v.            Matrix addition follows cancellation law,                                                                                                                                              ik      k              1j           i2     2j                       in     nj
                                                                      i.e. A + H = A + C  B = C (left cancellation law)                                                                                                                           k1
                                                                                                                                                                                                                                       and it is denoted by  C = AB.
                                                                      and B+A=C+ A B = C (right cancellation law)                                                                                             Transpose of a Matrix
                                                                      Note Two matrices are said to be conformable                                                                                                            If A =  [a ]                        is a matrix of order m × n, then the
                                                                      for addition or subtraction. if they are of the same                                                                                                                           ij   m×n
                                                                      order.                                                                                                                                                  transpose of A can be obtained by changing all
                                                        2.            Subtraction of Two Matrices                                                                                                                             rows to columns and all columns to rows
                                                                                                                                                                                                                              i.e. transpose of A =[a ]n × m
                                                                      Let A = [a ]                           and B = [b ]                                are two matrices                                                                                                             Tji
                                                                                                 ij  m×n                                   ij    m×n                                                                          It is denoted by A', A  or At.
                                                                      of same  order. Then, A – B = C = [C ]                                                                          ,
                                                                                                                                                                           ij  mxn                                                                                  1          4
                                                                      where c  = a  – b ,                                                                                                                                                                                        
                                                                                           ij           ij          ij                                                                                                                                                                                                     1         2 3
                                                                      Example 2                                                                                                                                                                                      2 5                                                                           
                                                                                                                                                                                                                                                                                                               T 
                                                                                                                                                                                                                              e.g. Let   A =                                                , then AA = 4                             5 6
                                                                                                                                                                                                                                                                  3          6                                                                   23
                                                                                             2 4                                             1          3                                                                                                                        32
                                                                      If A =                               and B =                                       , then A – B is
                                                                                                                                                         
                                                                                             3 2                                           2 5                                                                               Properties of Transpose of a Matrix
                                                                                                                                                         
                                                                                   3 7                                                         1          1                                                                   If A and B are two matrices and k is a scalar, then
                                                                      a.                                                        b.                                                                                        i.       (A')' = A                                          ii. (A + B)' = A' + B'
                                                                                                                                                            
                                                                                   5 7                                                         5 3                                                                           iii: (kA)' = kA'                                            iv.(AB)'=B'A' (reversal law)
                                                                                                                                                            
                                                        MATHEMATICS – MHT-CET                                                                                                                                                                               Himalaya Publication Pvt. Ltd.
                       DGT Group - Tuitions (Feed Concepts)   XIth – XIIth | JEE | CET | NEET  |  Call : 9920154035 / 8169861448
                                                                                                                                                                                                                                                       4
                                                                         DGT MH –CET 12th MATHEMATICS  Study Material
                                                                                                                            Matrices                                                                                    21
                                             Note If A. Band C are any three matrices                                                          If A and B are idempotent matrices, then A +B is
                                             conformable for multiplication. then (ABC)' =                                                     an idempotent, if AB = – BA.
                                             C'B' A'.                                                                                          ii. Nilpotent Matrix A square matrix A is called
                                             Conjugate of a Matrix                                                                                   nilpotent matrix, if it satisfies the relation
                                             The matrix obtained from any given matrix A                                                       Ak = 0 and Ak–1  0.
                                             containing complex numbers as its elements, on                                                    where, k is a positive integer.
                                             replacing its elements by the corresponding                                                       iii. Involutory Matrix A square matrix A is called
                                             conjugate complex numbers is called conjugate                                                           an involutory matrix, if it satisfies the relation
                                             of A and is denoted by A                                                                          iv. Symmetric Matrix A square matrix A is called
                                                                                                                                                     symmetric matrix, if it satisfies the relation
                                                                   i2i          23i                                                              A'=A
                                             e.g.  if A =   4 5i                56i, then                                                  If A and B are symmetric matrices of the same
                                                                                                                                                   order, then
                                                          12i 23i                                                                            (a) AB is symmetric if AB = BA.
                                                                                
                                                    A 45i 56i                                                                              (b)A ± B, AB + BA are also symmetric matrices.
                                                                                                                                             If A is symmetric matrix, then A-I will also be
                                             Properties of Conjugate of a Matrix                                                                     symmetric matrix.
                                             If A and B are two matrices, then                                                                 v. Skew-symmetric Matrix
                                             i.    (A)A                            ii.   (AB)AB                                             A square matrix A is called skew-symmetric matrix,
                                             iii.                                   iv.                                                        if it satisfies the relation
                                                    ABA.B                                (kA)kA,kisareal scalar                                    A' = – A
                                    Conjugate Transpose of a Matrix.                                                                           If A and B are two skew-symmetric matrices, then
                                             The transpose of the conjugate of a matrix A is                                                   (a) A + B, AB – BA are skew-symmetric matrices.
                                             called conjugate transpose of A and is denoted by                                                 (b)AB + BA is a symmetric matrix.
                                             A or A.                                                                                          Determinant of skew-symmetric matrix of odd
                                              A– = Conjugate of A' = (A)                                                                       order is zero.
                                             Note The transpose of the conjugate of A is the                                                   Note Every square matrix can be uniquely
                                             same as the conjugate of the transpose of A                                                       expressed as the sum of symmetric and skew-
                                                                                                                                               symmetric matrix.
                                                                                                                                               i.e.A =  1  (A+A') +  1  (A–A')
                                                                  24i              3         59i                                                         2                     2
                                                                      4         i             3i 
                                             e.g.  If A =                                               
                                                                      2            5          4i                                                          1                                                 1
                                                                                                                                             where.             (A + A') is symmetric and                         (A – A')
                                                                 24i             4            2                                                              2                                                 2
                                                                                                                                             is  skew-symmetric.
                                                                   3         i            5                                     Example 3
                                             then A– =                                             
                                                               59i              3i        4i
                                                                                                                                                           6 9                        2 6 0
                                             Properties of Conjugate Transpose ofa                                                             If A =    2          3  and B = 7               9 8, then AB is
                                                                                                                                                                                                        
                                             Matrix                                                                                                   75 25 117                           75 117 72
                                             i.    For a matrix A, (A') = CAY                                                                  a.                                   b.                            
                                             ii.   (A–)– = A                                                                                          72 39 24                            25 39              24
                                             iii. If A and B are two matrices, then                                                                    72 29 24
                                                   (A+B)– = A– + B–                                                                            c.                                   d. Not defined
                                                           e          e                                                                               75 25 117
                                             iv. (kA)  = kA , where k is any real scalar.                                                                                    
                                                            –       –     –
                                             v. (AB)  = B  A                                                                                   Sol (b) The matrix A has 2 columns which is
                                             Special Types of Matrices                                                                         equal to the number of rows of B. Hence, AB is
                                             i.    Idempotent Matrix A square matrix A is called                                               defined.
                                                   an idempotent matrix, if it satisfies the relation
                                                   A2 = A.
                                    MATHEMATICS – MHT-CET                                                                                                         Himalaya Publication Pvt. Ltd.
               DGT Group - Tuitions (Feed Concepts)   XIth – XIIth | JEE | CET | NEET  |  Call : 9920154035 / 8169861448
The words contained in this file might help you see if this file matches what you are looking for:

...Dgt mh cet th mathematics study material matrices iv square matrix amatrix in which number of syllabus rows is equal to the columns called a elements types algebra ij m n e g and am amn are two null order where entry at ith row andjth column respectively orderofamatrixaismx wheremis vi diagonal if all its non zero mayor may i has only one not be any nuymber d principal x then denoted as diag ii said it b c by note zeroes iii rectangular lie between an or vice versa vii triple except possibly for those lying on immediately above well below mht himalaya publication pvt ltd group tuitions feed concepts xith xiith jee neet call known sub constituted array left after deleting some both submatrix...

no reviews yet
Please Login to review.