Find non-zero values of x satisfying the matrix equation $x\begin{bmatrix}2x&2\\3&x\end{bmatrix}+2\begin{bmatrix}8&5x\\4&4x\end{bmatrix}=2\begin{bmatrix}x^2+8&24\\10&6x\end{bmatrix}$.

4

Given that

$x\begin{bmatrix}2x&2\\3&x\end{bmatrix}+2\begin{bmatrix}8&5x\\4&4x\end{bmatrix}=2\begin{bmatrix}x^2+8&24\\10&6x\end{bmatrix}$

$⇒\begin{bmatrix}2x^2&2x\\3x&x^2\end{bmatrix}+\begin{bmatrix}16&10x\\8&8x\end{bmatrix}=\begin{bmatrix}2x^2+16&47\\20&12x\end{bmatrix}$

$⇒\begin{bmatrix}2x^2+16&2x+10x\\3x+8&x^2+8x\end{bmatrix}=\begin{bmatrix}2x^2+16&47\\20&12x\end{bmatrix}$

Comparing the elements, we get

$2x+10x=48$

$⇒12x=48$

$⇒x=4$

This value of x also satisfies the equations $3x+8=20$ and $x^2 + 8x = 12x$.