Question

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

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

Answer 1

Answer: (B)

We know that, $(\operatorname{adj} A) A=|A| I=\begin{bmatrix}|A| & 0 & 0 \\ 0 & |A| & 0 \\ 0 & 0 & |A|\end{bmatrix}$
Writing determinants of matrices on both sides, we have
$|(\operatorname{adj} A) A|=\begin{vmatrix} |A| & 0 & 0 \\0 & |A| & 0 \\0 & 0 & |A|\end{vmatrix}$
i.e., $|(\operatorname{adj} A)||A|=|A|^3\begin{vmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{vmatrix}$
i.e., $|(\operatorname{adj} A)||A|=|A|^3(1)$
i.e., $|(\operatorname{adj} A)|=|A|^2 (\because|A| \neq 0)$