Question

If $A$ is a square matrix of order 3, then

• A. $|adjA|=|A{|}^2$
• B. $A\left(adjA\right)=|A|I_3$
• C. $A$ is invertible if $|A|\ne 0$
• D. none of these

Answer 1

Answer: (A), (B), (C)

$|adjA|=|A{|}^2$
$A$ is invertible if $|A|\ne 0$
$A\left(adjA\right)=|A|I_3$

$A$ is a square matrix of order $3.$

$|adjA|=|A{|}^{n-1}=|A{|}^2$

we know $A^{-1}=\frac{1}{|A|}adjA$

$\Rightarrow A{A}^{-1}=\frac{1}{|A|}A\left(adjA\right)$

$\therefore A\left(adjA\right)=|A|I_3$

$A$ is invertible iff it is a non-singular matrix.

i.e $|A|\ne 0$

Hence, options $A,B,$ and $C.$