Question

If $A$ is a non-singular square matrix of order 3 , then

• A. $|(\mathrm{adj}A)|=|A{|}^3$
• B. $|(\mathrm{adj}A)|=|A{|}^2$
• C. $|(\mathrm{adj}A)|=|A|$
• D. $|(\mathrm{adj}A)|=2|A|$

Answer 1

Answer: (B)

We know that, $(\mathrm{adj}A)A=|A|I=\left[\begin{array}{ccc}|A|& 0& 0\\ 0& |A|& 0\\ 0& 0& |A|\end{array}\right]$
Writing determinants of matrices on both sides, we have
$|(\mathrm{adj}A)A|=\left|\begin{array}{ccc}|A|& 0& 0\\ 0& |A|& 0\\ 0& 0& |A|\end{array}\right|$
i.e., $|(\mathrm{adj}A)||A|=|A{|}^3\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$
i.e., $|(\mathrm{adj}A)||A|=|A{|}^3(1)$
i.e., $|(\mathrm{adj}A)|=|A{|}^2(\because |A|\ne 0)$