Question

$S_1:$ For a $3\ast 3$ square matrix A if adj(adj A) = (det A) A then det (adj A) =${\left(detA\right)}^2$
$S_2:$ For a $3\ast 3$ square matrix a if det (2A) = 8(det A) then det $adj\left(2A\right)=8|adjA|$

• A. Only $S_1$ is true
• B. Only $S_2$ is true
• C. both $S_1$ and $S_2$ are true
• D. Neither $S_1$ nor $S_2$ is true

Answer 1

The correct option is A Only $S_1$ is true

$\left|adjA\right|={\left|A\right|}^{n-1}always$
the $s_1$ is true
s=2
for a $3\times 3$ matrix
first part is always correct as
$\left\{2A\right\}=2^n\left|A\right|$
second part
$\left(adj2A\right)={\left|2A\right|}^{n-1}$
$\Rightarrow \left|adj2A\right|={\left|2A\right|}^2$
$=\left|2^3{\left|A1\right|}^2\right|$
as $\left|kA\right|=k^n\left|A\right|$
thus, $\left|adj2A\right|=2^6{\left|A\right|}^2$
but ${\left|A\right|}^2=\left|adjA\right|$
thus, $\left|adj2A\right|=2^6\left|adjA\right|$
hence, $s_2$ si false.