Question

If matrix $A=\left[\begin{array}{ccc}2& 2& 3\\ 1& -2& 3\\ x& 1& -1\end{array}\right]$ is singular then find $x$.

Answer 1

$A$ is singular$\Rightarrow \left|A\right|=0$

$\left|A\right|=\left|\begin{array}{ccc}2& 2& 3\\ 1& -2& 3\\ x& 1& -1\end{array}\right|=0$

$\Rightarrow 2(2-3)-2(-1-3x)+3(1+2x)=0$

$\Rightarrow 2(-1)+2(1+3x)+3(1+2x)=0$

$\Rightarrow -2+2+6x+3+6x=0$

$\Rightarrow 12x+3=0$

$\Rightarrow 12x=-3$

$\Rightarrow x=\frac{-3}{12}$

$\therefore x=\frac{-1}{4}$