Question

For $3\times 3$ matrices $A$ and $B,$ which of the following statements is (are) CORRECT?

• A. $AB$ is skew-symmetric if $A$ is symmetric and $B$ is skew-symmetric.
• B. $(adjA{)}^T=adj(A^T)$ for all invertible matrix $A.$
• C. $AB+BA$ is symmetric for all symmetric matrices $A$ and $B.$
• D. $(adjA{)}^{-1}=adj(A^{-1})$ for all invertible matrix $A.$

Answer 1

Answer: (B), (C), (D)

$(adjA{)}^{-1}=adj(A^{-1})$ for all invertible matrix $A.$

$A$ is symmetric and $B$ is skew-symmetric.
Let $P=AB$
Now, $P^T=(AB{)}^T=B^T{A}^T=-BA$
$\Rightarrow P^T\ne -P$
$\therefore AB$ is not skew symmetric.

We know that
$A\left(adjA\right)=|A|I_3$
$\Rightarrow \left(A\right(adjA){)}^T=(|A|I_3{)}^T$
$\Rightarrow (adjA{)}^T{A}^T=|A|I_3$
$\Rightarrow (adjA{)}^T=|A^T|(A^T{)}^{-1}$
$\therefore (adjA{)}^T=adj(A^T)$ for all invertible matrix $A.$

Let $P=AB+BA$
$\therefore P^T=(AB+BA{)}^T=(AB{)}^T+(BA{)}^T$
$=B^T{A}^T+A^T{B}^T=BA+AB$
$=AB+BA=P$
$\therefore AB+BA$ is symmetric for all symmetric matrices $A$ and $B.$

$adj\left(A^{-1}\right)=\frac{A}{|A|}\cdots \left(1\right)$
Also, $A\left(adjA\right)=|A|I_3$
$\Rightarrow \left(adjA\right)=|A|A^{-1}$
$\Rightarrow (adjA{)}^{-1}=\frac{A}{|A|}\cdots (2)$
From $\left(1\right)$ and $\left(2\right),$ we get
$(adjA{)}^{-1}=adj(A^{-1})$