In the matrix A = a 1 x 2 3 x 2 - y 0 5 - 2 5 , write elements a 23 , a 31 and a 12 .

We have, A = a 1 x 2 3 x 2 - y 0 5 - 2 5 Since, we know that a i j is a representation of element lying in the i th row and j th column. a 23 corresponds to 2 nd row and 3 rd column, ⇒ a 23 = x 2 - y a 31 corresponds to 3 rd row and 1 st column, ⇒ a 31 = 0 a 12 corresponds to 1 st row and 2 nd column, ⇒ a 12 = 1 Hence, a 23 = x 2 - y , a 31 = 0 , a 12 = 1 .

In the matrix A = a 1 x 2 3 x 2 - y 0 5 - 2 5 , write elements a 23 , a 31 and a 12 .

We have, A = a 1 x 2 3 x 2 - y 0 5 - 2 5 Since, we know that a i j is a representation of element lying in the i th row and j th column. a 23 corresponds to 2 nd row and 3 rd column, ⇒ a 23 = x 2 - y a 31 corresponds to 3 rd row and 1 st column, ⇒ a 31 = 0 a 12 corresponds to 1 st row and 2 nd column, ⇒ a 12 = 1 Hence, a 23 = x 2 - y , a 31 = 0 , a 12 = 1 .

Construct a 2 × 2 matrix where a i j = i - 2 j 2 2 .

Let A be the matrix of order 2 × 2 , where a i j = i - 2 j 2 2 Then, A = a 11 a 12 a 21 a 22 = 1 - 2 ( 1 ) 2 2 1 - 2 ( 2 ) 2 2 2 - 2 ( 1 ) 2 2 2 - 2 ( 2 ) 2 2 = 1 2 9 2 0 2 .

Construct 2 × 2 matrix where a i j = - 2 i + 3 j .

Let A be the matrix of order 2 × 2 where a i j = - 2 i + 3 j So, A = a 11 a 12 a 21 a 22 = - 2 1 + 3 1 - 2 1 + 3 2 - 2 2 + 3 1 - 2 2 + 3 2 = - 2 + 3 - 2 + 6 - 4 + 3 - 4 + 6 = 1 4 1 2 .

Construct a 3 × 2 matrix whose elements are given by a i j = e i x sin j x .

Let A be the matrix of order 3 × 2 where a i j = e i x sin j x . So, A = a 11 a 12 a 21 a 22 a 31 a 32 = e x sin x e x sin 2 x e 2 x sin x e 2 x sin 2 x e 3 x sin x e 3 x sin 2 x .

Find values of a and b if A = B , where A = a + 4 3 b 8 - 6 2 × 2 , B = 2 a + 2 b 2 + 2 8 b 2 - 5 b 2 × 2 .

We have A = a + 4 3 b 8 - 6 and B = 2 a + 2 b 2 + 2 8 b 2 - 5 b If A = B , then by equality of matrices we have: a + 4 = 2 a + 2 . . . . . . . i b 2 + 2 = 3 b . . . . . . . . i i b 2 - 5 b = - 6 . . . . . . . . i i i By solving ( i ) , we get a = 2 On solving Eq. ( i i ) and ( i i i ) ,we get ⇒ 2 + 5 b = 3 b + 6 ⇒ 2 b = 4 ⇒ b = 2 Hence, a = 2 and b = 2 .

If possible, find the sum of the matrices A and B , where A = 3 1 2 3 2 × 2 and B = x y z a b 6 2 × 3 .

The given matrices A = 3 1 2 3 is of order 2 × 2 and B = x y z a b 6 is of order 2 × 3 . Since, the matrices are not of same order. So, addition of two matrices is not possible. Hence, the sum of the matrices A and B is not possible.

If X = 3 1 - 1 5 - 2 - 3 2 × 3 and Y = 2 1 - 1 7 2 4 2 × 3 , find X + Y .

We have X = 3 1 - 1 5 - 2 - 3 and Y = 2 1 - 1 7 2 4 Addition of matrices X and Y is possible because both matrices are of same order 2 × 3 . ⇒ X + Y = 3 + 2 1 + 1 - 1 - 1 5 + 7 - 2 + 2 - 3 + 4 ⇒ X + Y = 5 2 - 2 12 0 1 . Hence, X + Y = 5 2 - 2 12 0 1 .

If X = 3 1 - 1 5 - 2 - 3 2 × 3 and Y = 2 1 - 1 7 2 4 2 × 3 , find 2 X - 3 Y .

We have X = 3 1 - 1 5 - 2 - 3 and Y = 2 1 - 1 7 2 4 ∴ 2 X = 2 3 1 - 1 5 - 2 - 3 = 6 2 - 2 10 - 4 - 6 And, 3 Y = 3 2 1 - 1 7 2 4 = 6 3 - 3 21 6 12 Subtraction of matrices 2 X and 3 Y is possible because both matrices are of same order 2 × 3 . So, 2 X - 3 Y = 6 - 6 2 - 3 - 2 + 3 10 - 21 - 4 - 6 - 6 - 12 ⇒ 2 X - 3 Y = 0 - 1 1 - 11 - 10 - 18 . Hence, 2 X - 3 Y = 0 - 1 1 - 11 - 10 - 18 .

If X = 3 1 - 1 5 - 2 - 3 2 × 3 and Y = 2 1 - 1 7 2 4 2 × 3 , find a matrix Z such that X + Y + Z is a zero matrix.

We have X = 3 1 - 1 5 - 2 - 3 and Y = 2 1 - 1 7 2 4 Addition of matrices X and Y is possible because both matrices are of same order 2 × 3 . The matrix Z should be of order 2 × 3 [ ∵ X and Y are of order 2 × 3 .] ⇒ Z = a b c d e f ⇒ X + Y + Z = 3 + 2 + a 1 + 1 + b - 1 - 1 + c 5 + 7 + d - 2 + 2 + e - 3 + 4 + f ⇒ X + Y + Z = 5 + a 2 + b c - 2 12 + d e 1 + f We know that X + Y + Z = 0 ⇒ 5 + a 2 + b c - 2 12 + d e 1 + f = 0 ⇒ 5 + a 2 + b c - 2 12 + d e 1 + f = 0 0 0 0 0 0 By equality of matrices, we get 5 + a = 0 ⇒ a = - 5 2 + b = 0 ⇒ b = - 2 c - 2 = 0 ⇒ c = 2 12 + d = 0 ⇒ d = - 12 and e = 0 and 1 + f = 0 ⇒ f = - 1 ∴ Z = - 5 - 2 2 - 12 0 - 1 .