Question:
For what value of $x$ is the matrix $\left[\begin{array}{ll}6-x & 4 \\ 3-x & 1\end{array}\right]$ singular?
Solution:
$\left|\begin{array}{ll}6-x & 4 \\ 3-x & 1\end{array}\right|$ is singular when its determinant is 0 .
$\Rightarrow\left|\begin{array}{cc}6-x & 4 \\ 3-x & 1\end{array}\right|=0$
$\Rightarrow(6-x)-4(3-x)=0$
$\Rightarrow 6-x-12+4 x=0$
$\Rightarrow 3 x-6=0$
$\Rightarrow 3 x=6$
$\Rightarrow x=\frac{6}{3}=2$
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