Question

If a matrix A is both symmetric and skew symmetric, then A is necessarily a ______.

Options

• Diagonal matrix

• Zero square matrix

• Square matrix

• Identity matrix

Answer 1

If a matrix A is both symmetric and skew symmetric, then A is necessarily a zero square matrix.

Explanation:

If matrix A is symmetric

A^T = A
If matrix A is skew-symmetric

A^T = –A

Also, diagonal elements are equal to zero.

Since, matrix A is both symmetric and skew-symmetric.

∴ A = A^T = –A

Which is only possible if A is zero matrix.

`A = [(0, 0),(0, 0)] = A^T = - A`

Thus, if a matrix A is both symmetric and skew symmetric, it must be zero.