Question

If $|AdjA|=81$, find $|A|$ if $A$ is a matrix of order $3\times 3$. Find also $|A^T|$ & $A^{-1}$

Answer 1

We know that
$|adjA|=|A{|}^{n-1}$ $n=3$
$|n{|}^2=81$
$\Rightarrow |A|=9$
$|A^{-1}|=\frac{1}{|A|}=\frac{1}{9}$
$|A|=-|A|=-9$