Question:
Let A be a nonsingular square matrix of order 3 × 3. Then 
is equal to
A. $|A|$
B. $|A|^{2}$
C. $|A|^{3}$
D. $3|A|$
Solution:
Answer: B
We know that,
$(\operatorname{adj} A) A=|A| I=\left[\begin{array}{ccc}|A| & 0 & 0 \\ 0 & |A| & 0 \\ 0 & 0 & |A|\end{array}\right]$
$\Rightarrow|(\operatorname{adj} A) A|=\left|\begin{array}{lll}A \mid & 0 & 0 \\ 0 & |A| & 0 \\ 0 & 0 & |A|\end{array}\right|$
$\left.\Rightarrow|\operatorname{adj} A| A|=| A\right|^{3}\left|\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right|=|A|^{3}(I)$
$\therefore|\operatorname{adj} A|=|A|^{2}$
Hence, the correct answer is B.
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