Question

Find value(s) of $x,$ if
$\left(i\right)$$\left|\begin{array}{cc}2& 4\\ 5& 1\end{array}\right|=\left|\begin{array}{cc}2x& 4\\ 6& x\end{array}\right|$
$\left(ii\right)$ $\left|\begin{array}{cc}2& 3\\ 4& 5\end{array}\right|=\left|\begin{array}{cc}x& 3\\ 2x& 5\end{array}\right|$

Answer 1

$\left(i\right)$ Given: $\left|\begin{array}{cc}2& 4\\ 5& 1\end{array}\right|=\left|\begin{array}{cc}2x& 4\\ 6& x\end{array}\right|$
$\Rightarrow 2\left(1\right)-5\left(4\right)=2x\left(x\right)-6\left(4\right)$
$\Rightarrow 2-20=2x^2-24$
$\Rightarrow -18=2x^2-24$
$\Rightarrow 2x^2=-18+24$
$\Rightarrow 2x^2=6$
$\Rightarrow x^2=\frac{6}{2}$
$\Rightarrow x=\pm \sqrt{3}$
Hence, values of $x$ are $\pm \sqrt{3}.$

$\left(ii\right)$ Given: $\left|\begin{array}{cc}2& 3\\ 4& 5\end{array}\right|=\left|\begin{array}{cc}x& 3\\ 2x& 5\end{array}\right|$
$\Rightarrow 2\left(5\right)-4\left(3\right)=x\left(5\right)-2x\left(3\right)$
$\Rightarrow 10-12=5x-6x$
$\Rightarrow -2=-x$
$\Rightarrow x=2$
Hence, value of $x$ is $2.$