Question

If $A$ is a square matrix, then

• A. $A+A^T$ is symmetric
• B. $A{A}^T$ is skew-symmetric
• C. $A^T+A$ is skew-symmetric
• D. $A^T$ is skew-symmetric

Answer 1

The correct option is A

$A+A^T$ is symmetric

Explanation for the correct option:

Given, $A$ is a square matrix

As we know,

$\begin{array}{rcl}{\left(A+A^T\right)}^T& =& A^T+{\left(A^T\right)}^T\\ & =& A^T+A\mathbf{[}\mathbf{\because }\mathbf{}{\left(A^T\right)}^{\mathbf{T}}\mathbf{=}\mathbf{}\mathit{A}\mathbf{}\mathbf{]}\end{array}$

$\therefore$$A+A^T$ is a symmetric matrix.

Hence, Option ‘A’ is Correct.