Question

Let $A$ and $B$ be two square matrices of order $3.$ Then which of the following statements is(are) CORRECT?

• A. $AB{A}^T$ is symmetric matrix.
• B. $AB-BA$ is skew symmetric matrix.
• C. If $B=|A|A^{-1},$ $|A|\ne 0,$ then $adj\left(A^T\right)-B$ is skew symmetric matrix.
• D. If $B+A^T=O$ and $A$ is skew symmetric matrix, then $B^{15}$ is also skew symmetric matrix.

Answer 1

Answer: (C), (D)

If $B+A^T=O$ and $A$ is skew symmetric matrix, then $B^{15}$ is also skew symmetric matrix.

${\left(AB{A}^T\right)}^T=A{B}^T{A}^T$
$\therefore AB{A}^T$ is not symmetric.


$(AB-BA{)}^T=(AB{)}^T-(BA{)}^T$
$\Rightarrow (AB-BA{)}^T=B^T{A}^T-A^T{B}^T$
$\therefore AB-BA$ is not symmetric.


$B=|A|\cdot \frac{adj\left(A\right)}{|A|}(\because A^{-1}=\frac{adj\left(A\right)}{|A|})$
$\Rightarrow B=adj\left(A\right)$
Now, ${\left(adj\right(A^T)-B)}^T$
$=\left(\right(adjA{)}^T-adjA{)}^T(\because adj{A}^T=(adjA{)}^T)$
$=-\left(\right(adjA{)}^T-B)$
$\therefore adj\left(A^T\right)-B$ is skew symmetric.


$B+A^T=O$
$\Rightarrow B=A(\because A^T=-A)$
$\Rightarrow {\left(B^{15}\right)}^T={\left(A^{15}\right)}^T$
$\Rightarrow {\left(B^{15}\right)}^T={\left(A^T\right)}^{15}=-A^{15}$
$\Rightarrow {\left(B^{15}\right)}^T=-B^{15}$
$\therefore B^{15}$ is skew symmetric.