Question

If $A$ is a square matrix of order $m\times n$, then $adj\left(adjA\right)$ is equal to

• A. $|A{|}^{n}A$
• B. $|A{|}^{n-1}A$
• C. $|A{|}^{n-2}A$
• D. $|A{|}^{n-3}A$

Answer 1

The correct option is C $|A{|}^{n-2}A$

$(adjA{)}^{-1}=\frac{adj\left(adjA\right)}{|adjA|}$
$\Rightarrow (|A|A^{-}{)}^{-1}=\frac{adj\left(adjA\right)}{|adjA|}$
$\Rightarrow \frac{A}{|A|}=\frac{adj\left(adjA\right)}{|adjA|}$
$\Rightarrow adj\left(adjA\right)=\frac{|adjA|A}{|A|}$
$\therefore adj\left(adjA\right)=|A{|}^{n-2}A$
Hence, option C.