Question

If A is an invertible matrix of order $3\times 3$ such that $\left|A\right|=5$, find the value of $\left|A^{-1}\right|$ .

Answer 1

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

We Know, $A.A^{-1}=I$

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

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

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

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

$|adj\left(A\right)|=|A^2|$

$=5^2=25$

$A^{-1}=\frac{25}{5}=5$