Let A be a nonsingular square matrix of order 3 × 3. Then is equal to
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:

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.