Question

If A is an n x n non-singular matrix, then $\left|AdjA\right|$ is:

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

Answer 1

The correct option is C ${\left|A\right|}^{n-1}$

$A.adj.A=\left|A\right|$
$\left|A\right(adj.A\left)\right|={\left|\begin{array}{cccc}\left|A\right|& 0& 0& 0\\ .& \left|A\right|& .& .\\ .& .& \left|A\right|& .\\ 0& .& .& \left|A\right|\end{array}\right|}_{n\times n}$
$\left|A\right||adj.A|={\left|A\right|}^n$
$\therefore |adj.A|={\left|A\right|}^{n-1}$