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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 33
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
31 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.
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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
32
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 ] i1,2....,mand j1,2,....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
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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. 21 43 1 1
= =
Therefore, addition of A and B is defined and is
3(2) 25 5 3
given by
3. Scalar Multiplication
Let A = [a ] be any m×n matrix and k be any
ij
2 3 1 5 11 scalar. Then, the matrix obtained by multiplying
A + B = 1 each element of A by k is called the scalar
22 33 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) k1
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 23
2 4 1 3 32
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)
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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
i2i 23i A'=A
e.g. if A = 4 5i 56i, then If A and B are symmetric matrices of the same
order, then
12i 23i (a) AB is symmetric if AB = BA.
A 45i 56i (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. (AB)AB A square matrix A is called skew-symmetric matrix,
iii. iv. if it satisfies the relation
ABA.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')
24i 3 59i 2 2
4 i 3i
e.g. If A =
2 5 4i 1 1
where. (A + A') is symmetric and (A – A')
24i 4 2 2 2
is skew-symmetric.
3 i 5 Example 3
then A– =
59i 3i 4i
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.
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