Question

Let A be the non-singular square matrix of order $3\times 3$ then |adj A| is equal to

a) $|A|$
b) $|A{|}^2$
c) $|A{|}^3$
d) $3|A|$

Answer 1

We know that $\left(adjA\right)A=|A|I$
$=|A|\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}|A|& 0& 0\\ 0& |A|& 0\\ 0& 0& |A|\end{array}\right]\Rightarrow |\left(adjA\right)A|=\left[\begin{array}{ccc}|A|& 0& 0\\ 0& |A|& 0\\ 0& 0& |A|\end{array}\right]\Rightarrow |\left(adjA\right)A|=|A{|}^3\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|=|A{|}^3|I|\Rightarrow |adjA||A|=|A{|}^3(\because (I|=1\left)\right)\therefore |\left(adjA\right)|=|A{|}^2$
Hence, the correct option is (b)