Question

How many of the following statement are true ?
$S_1$: If $B$ is symmetric matrix then $AB{A}^T$ is symmetric
$S_2$ : If $A^4$ is singular matrix then $A$ is also singular
$S_3$ : If $AB=O$ and $|A|$ is non zero then $B$ must be a null matrix
$S_4$ : If $|A|\ne 0$ and $\left(adjA\right)B\ne 0$ then matrix equation $AX=B$ has no solution

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$S_1$: If $B$ is symmetric matrix then $AB{A}^T$ is symmetric

symmetric$\Rightarrow B=B^T$

$(AB{A}^T{)}^T=(A^T{)}^T{B}^T{A}^T=AB{A}^T$

Therefore, $S_1$ is true.

$S_2$ : If $A^4$ is singular matrix then $A$ is also singular

Given, $\left|A^4\right|=0\left(singular\right)$

$\left|AB\right|=\left|A\right|\left|B\right|\Rightarrow \left|A^4\right|={\left|A\right|}^4=0\Rightarrow \left|A\right|=0$

Therefore, $A$ is also singular.

Therefore, $S_2$ is true.

$S_3$ : If $AB=O$ and $|A|$ is non zero then $B$ must be a null matrix

$AB=0,\left|A\right|\ne 0$

So, $A^{-1}exists$

Multiply by $A^{-1}$on both sides.

$A^{-1}AB=B=0$

Therefore, $S_3$ is true.

$S_4$ : If $|A|\ne 0$ and $\left(adjA\right)B\ne 0$ then matrix equation $AX=B$ has no solution

$\left|A\right|\ne 0$

So, $A^{-1}exists$

$AX=B\Rightarrow X=A^{-1}B,X$ has unique solution.

Therefore, $S_4$ is false.