Question

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

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

Answer 1

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

$A^{-1}=\frac{AdjA}{detA}=\frac{I}{A}=\frac{AdjA}{\left|A\right|}$

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

Replacing $A=adjA$

$adjA\left(AdjAdjA\right)=\left|AdjA\right|$

$\Rightarrow Adj\left(AdjA\right)=\frac{\left|Adj\right(A\left)\right|}{\left|A\right|}\times A$

$=\frac{{\left|A\right|}^{n-1}}{\left|A\right|}\times A$

$={\left|A\right|}^{n-2}\times A$

$Adj\left(AdjA\right)={\left|A\right|}^{n-2}\times 2$

$n\to$order

Option C is correct