Question

lf the adjoint of a $3\times 3$ matrix $P$ is $\left[\begin{array}{ccc}1& 4& 4\\ 2& 1& 7\\ 1& 1& 3\end{array}\right]$, then the possible value(s) of the determinant of $P$ is (are):

• A. $-2$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

Answer: (A), (D)

$2$
$-2$

$Adj.P=\left|\begin{array}{ccc}1& 4& 4\\ 2& 1& 7\\ 1& 1& 3\end{array}\right|$

$\Rightarrow \mid Adj.P\mid =-4+4+4$

$=4$

$\mid Adj.P\mid =\mid P{\mid }^2[\because \mid AdjP\mid =\mid P{\mid }^{n-1}]$

$\Rightarrow \mid P{\mid }^2=4$

$\Rightarrow P=\pm 2$