Question

If a $n\times n$ square matrix $A$ is such that $\left|A\right|=D$ and $\left|adjA\right|=D^{\prime }$, then-

• A. $D{D}^{\prime }=D^n$
• B. $D{D}^{\prime }=D^{n-1}$
• C. $D{D}^{\prime }=D^2$
• D. none of these

Answer 1

Answer: (A)

$D{D}^{\prime }=D^n$

Considering that $A$ is an nth order square matrix, then
$|adjA|=|A{|}^{n-1}$
Now it is given that
$|adjA|=D^{\prime }$ and $|A|=D$.
Hence
$D^{\prime }=D^{n-1}$
Or
$D^{\prime }=\frac{D^n}{D}$
Or
$D.D^{\prime }=D^n$