Question

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

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (B)

We have,
$(\text{adj}(A){)}^T=\left[\begin{array}{ccc}1& -2& 1\\ 4& -5& -2\\ -2& 4& 1\end{array}\right]$
$\Rightarrow \text{adj}(A)={\left[\begin{array}{ccc}1& -2& 1\\ 4& -5& -2\\ -2& 4& 1\end{array}\right]}^T=\left[\begin{array}{ccc}1& 4& -2\\ -2& -5& 4\\ 1& -2& 1\end{array}\right]$
$|adjA|=\left|\begin{array}{ccc}1& 4& -2\\ -2& -5& 4\\ 1& -2& 1\end{array}\right|$
$=1(-5+8)-4(-2-4)-2(+4+5)$
$=3+24-18=9$
Now, we know that
$|\text{adj}A|=|A{|}^{n-1}$
$\Rightarrow |A{|}^{3-1}=9$
$\Rightarrow |A{|}^2=9$
$\Rightarrow |A|=\pm 3$