Construct a 2 × 2 matrix A = a i j , whose elements are given by a i j = i j .

To construct: A 2 × 2 matrix, A = a i j , whose elements are given by: a i j = i j Here a i j = i j So, the elements of matrix are: a 11 = 1 1 = 1 a 12 = 1 2 a 21 = 2 1 = 2 a 22 = 2 2 = 1 So, A = a 11 a 12 a 21 a 22 Therefore, A = 1 1 2 2 1

Construct a 2 × 2 matrix A = a i j , whose elements are given by a i j = ( i + j ) 2 2 ;

To construct: A 2 × 2 matrix, A = a i j whose elements are given by: a i j = ( i + j ) 2 2 . Thus, A = a 11 a 12 a 21 a 22 Here, a i j = ( i + j ) 2 2 , Therefore, a 11 = ( 1 + 1 ) 2 2 = 2 a 12 = ( 1 + 2 ) 2 2 = 9 2 a 21 = ( 2 + 1 ) 2 2 = 9 2 a 22 = ( 2 + 2 ) 2 2 = 8 Therefore, matrix A = 2 9 2 9 2 8

Construct a 3 × 2 matrix whose elements are given by a ij = 1 2 [ i − 3 j ] .

In general 3 × 2 matrix is given by A = a 11 a 12 a 21 a 22 a 31 a 32 Now a ij = 1 2 | i - 3 j | , i = 1 , 2 , 3 j = 1 , 2 Therfore a 11 = 1 2 | 1 - 3 × 1 | = 1 , a 12 = 1 2 ∣ 1 - 3 × 21 = 5 2 a 21 = 1 2 2 - 3 × 1 = 1 2 , a 22 = 1 2 ∣ 2 - 3 × 2 I = 2 a 31 = 1 2 | 3 - 3 × 1 | = 0 , a 32 = 1 2 | 3 - 3 × 2 | = 3 2 Hence the required matrix is given by A = 1 5 2 1 2 2 0 3 2

If matrix A = a i j 3 × 2 and a i j = 3 i - 2 j 2 of matrix A , then find A .

Given that A = a i j 3 × 2 ⇒ A = a 11 a 12 a 21 a 22 a 31 a 32 3 × 2 And also given that a i j = 3 i - 2 j 2 Now, a 11 = 3 1 - 2 1 2 = 3 - 2 2 = 1 a 12 = 3 1 - 2 2 2 = 3 - 4 2 = 1 a 21 = 3 2 - 2 1 2 = 6 - 2 2 = 16 a 22 = 3 2 - 2 2 2 = 6 - 4 2 = 4 a 31 = 3 3 - 2 1 2 = 9 - 2 2 = 49 a 32 = 3 3 - 2 2 2 = 9 - 4 2 = 25 Therefore, A = 1 1 16 4 49 25 3 × 2 .

Construct a ( 2 × 3 ) matrix whose elements are given by a i j = 2 i + j .

We know that a 2 × 3 matrix is a matrix having 2 rows, and 3 columns. Let A = a 11 a 12 a 13 a 21 a 22 a 23 , where a i j = 2 i + j , where i = 1 , 2 , and j = 1 , 2 , 3 . a 11 = 2 1 + 1 = 2 + 1 = 3 a 12 = 2 1 + 3 = 5 a 13 = 2 1 + 3 = 5 a 21 = 2 2 + 1 = 5 a 22 = 2 2 + 2 = 6 a 23 = 2 2 + 3 = 4 + 3 = 7 Hence, A = 3 5 5 5 6 7

Construct a 2 × 2 matrix whose elements are given by, a i j = | 2 i - 3 j | 3

Let A be the required matrix. It is a 2 × 2 matrix. So it has 2 rows and 2 columns. Given a i j = | 2 i - 3 j | 3 . So, a 11 = ( 2 × 1 ) - 3 × 1 3 = 1 3 a 12 = ( 2 × 1 ) - 3 × 2 3 = 4 3 a 21 = ( 2 × 2 ) - 3 × 1 3 = 1 3 a 22 = ( 2 × 2 ) - 3 × 2 3 = 2 3 In general, a matrix of 2 × 2 order given by A = a 11 a 12 a 21 a 22 . So, the matrix is A = 1 3 4 3 1 3 2 3 .

Let A = a i j 2 × 3 ; where a i j = i + j .construct A .

Given matrix A is of order 2 × 3 ∴ A = a 11 a 12 a 13 a 21 a 22 a 23 Given a i j = i + j Therefore, a 11 = 1 + 1 = 2 a 12 = 1 + 2 = 3 a 13 = 1 + 3 = 4 a 21 = 2 + 1 = 3 a 22 = 2 + 2 = 4 a 23 = 2 + 3 = 5 Hence, A = 2 3 4 3 4 5