Question

Let A be a square matrix. Which of the following is/are not skew-symmetric matrix/ces?

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

Answer 1

The correct option is C $A{A}^T-A^{T}A$

A. Let $Q=A-A^T$.
Then, $Q^T=(A-A^T{)}^T=A^T-(A^T{)}^T[\because (A+B{)}^T=A^T+B^T]$
$\Rightarrow Q^T=A^T-A$
$\Rightarrow Q^T=-(A-A^T)=-Q$
Therefore, Q is skew-symmetric.
B. similarly $A^T-A$ is skew symmetric matrix.
C. Let $Q=A{A}^T-A^{T}A$
Then $Q^T=(A{A}^T-A^{T}A{)}^T=(A{A}^T)T-(A^{T}A{)}^T=A{A}^T-A^{T}A=Q$
Hence $Q$ is symmetric matrix.
D. Here given $A$ is skew symmetric matrix $\Rightarrow A=-A^T$
$\Rightarrow A+A^T=0$ Hence $A+A^T$ is skew symmetric matrix. since zero matrix is skew symmetric matrix as well as symmetric matrix.
Hence option 'C' is the correct choice.