Question

If the matrix $A$ is both symmetric and skew symmetric, then

• A. $A$ is a diagonal matrix
• B. $A$ is a zero matrix
• C. $A$ is a square matrix
• D. None of these

Answer 1

Answer: (B)

$A$ is a zero matrix

If $A$ is both symmetric and skew symmetric, then we should have
$A^{\prime }=A$ and $A^{\prime }=-A$
$\Rightarrow A=-A$
$\Rightarrow A+A=0$
$\Rightarrow 2A=0$
$\Rightarrow A=0$
Therefore, $A$ is a zero matrix.