Question

Given a square matrix $A$ of order $3$, such that $|A^{\prime }|=15$, find $|A.adjA|$.

Answer 1

Given matrix $A$ of order $n,A^{-1}=\frac{adj\left(A\right)}{|A|}$

We know that, $A\ast A^{-1}=I$

$A\ast \frac{adj\left(A\right)}{|A|}=I$

$A\ast adj\left(A\right)=|A|\ast I$

$|A\cdot adj\left(A\right)|={|A|}^n$

$|A|\cdot |adj\left(A\right)|={|A|}^n$

$|adj\left(A\right)|={|A|}^{n=1}$

Therefore, here $|adj\left(A\right)|=5^2=25$