For what value of x , is the matrix A = 0 1 - 2 - 1 0 3 x - 3 0 a skew-symmetric matrix?

Given A = 0 1 - 2 - 1 0 3 x - 3 0 The transpose of a matrix is obtained by interchanging rows into columns or columns to rows. A T = 0 - 1 x 1 0 - 3 - 2 3 0 Since, A is a skew-symmetric matrix. ∴ A T = - A ⇒ 0 - 1 x 1 0 - 3 - 2 3 0 = - 0 1 - 2 - 1 0 3 x - 3 0 ⇒ 0 - 1 x 1 0 - 3 - 2 3 0 = 0 - 1 2 1 0 - 3 - x 3 0 Corresponding elements of equal matrices are equal. ⇒ x = 2 .

For what value of x , is the matrix A = 0 1 - 2 - 1 0 3 x - 3 0 a skew-symmetric matrix?

Given A = 0 1 - 2 - 1 0 3 x - 3 0 The transpose of a matrix is obtained by interchanging rows into columns or columns to rows. A T = 0 - 1 x 1 0 - 3 - 2 3 0 Since, A is a skew-symmetric matrix. ∴ A T = - A ⇒ 0 - 1 x 1 0 - 3 - 2 3 0 = - 0 1 - 2 - 1 0 3 x - 3 0 ⇒ 0 - 1 x 1 0 - 3 - 2 3 0 = 0 - 1 2 1 0 - 3 - x 3 0 Corresponding elements of equal matrices are equal. ⇒ x = 2 .

Write a 2 × 2 matrix which is both symmetric and skew-symmetric.

Let A = 0 0 0 0 A T = 0 0 0 0 Since A T = A , A is a symmetric matrix. Now, - A = - 0 0 0 0 ⇒ - A = 0 0 0 0 Since A T = - A , A is a skew-symmetric matrix. Thus, A = 0 0 0 0 is a null matrix is an example of a matrix that is both symmetric and skew - symmetric.

Construct a 2 × 2 matrix A = [ a i j ] whose elements a i j are given by a i j = | - 3 i + j | 2 , if i ≠ j ( i + j ) 2 , if i = j

Given a 2 × 2 matrix A = a i j a i j = | - 3 i + j | 2 , if i ≠ j ( i + j ) 2 , if i = j a 11 = 1 + 1 2 = 4 a 12 = - 3 × 1 + 2 2 = 1 2 a 21 = - 3 × 2 + 1 2 = 5 2 a 22 = 2 + 2 2 = 16 Hence, the matrix A = 4 1 2 5 2 16 .