Question

If $A$ is a $3 \times 3$ matrix and the matrix obtained by replacing the elements of $A$ with their corresponding cofactors is
$\begin{bmatrix}1 & -2 & 1 \\ 4 & -5 & -2 \\ -2 & 4 & 1\end{bmatrix}$ then a possible value of the determinant of $A$ is

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (B)

We have,
$(\text{adj}(A))^{T}=\begin{bmatrix}1 & -2 & 1 \\4 & -5 & -2 \\-2 & 4 & 1 \end{bmatrix}$
$\Rightarrow \text{adj}(A)=\begin{bmatrix}1 & -2 & 1 \\ 4 & -5 & -2 \\ -2 & 4 & 1\end{bmatrix}^{T}=\begin{bmatrix}1 & 4 & -2 \\ -2 & -5 & 4 \\ 1 & -2 & 1\end{bmatrix}$
$|a d j A|=\begin{vmatrix}1 & 4 & -2 \\ -2 & -5 & 4 \\ 1 & -2 & 1\end{vmatrix}$
$=1(-5+8)-4(-2-4)-2(+4+5)$
$=3+24-18=9$
Now, we know that
$|\text{adj} A|=|A|^{n-1}$
$\Rightarrow |A|^{3-1}=9 $
$\Rightarrow |A|^{2}=9 $
$\Rightarrow |A|=\pm 3$