Question

If $P=\left[\begin{array}{ccc}1& c& 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the adjoint of a $3\times 3$ matrix $Q$ and $detQ=4$, then the value of $c$ is

Answer 1

Given, $P=\text{adj}Q$ and $detQ=4$
We know that,
$det\left(\text{adj}Q\right)=(detQ{)}^{n-1},$ here $n=3$
$\Rightarrow \left|P\right|={\left|Q\right|}^2$
$\Rightarrow \left|P\right|=16$
$\Rightarrow \left|\begin{array}{ccc}1& c& 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right|=16$
$\Rightarrow 1(12-12)-c(4-6)+3(4-6)=16$
$\Rightarrow 2c=22\Rightarrow c=11$