Question

State whether the following statement is True or False:
If $A$ is skew symmetric matrix, then $A^2$ is a symmetric matrix.

• A. False
• B. True

Answer 1

The correct option is B True

Given that $A$ is skew symmetric
$\Rightarrow A^{\prime }=-A$ .... (1)
Now finding transpose of $A^2$
$\Rightarrow {\left(A^2\right)}^{\prime }=2{\left(A^{\prime }\right)}^2$
$[\because {\left(A^n\right)}^{\prime }=2{\left(A^{\prime }\right)}^n]$
$\Rightarrow {\left(A^2\right)}^{\prime }={(-A)}^2$ [Using $\left(1\right)$]
$\Rightarrow {\left(A^2\right)}^{\prime }={\left(A\right)}^2$
[If $A$ is symmetric matrix, then $A^{\prime }=A$]
$\Rightarrow A^2\text{is symmetric matrix}$.
Hence, the given statement is true.