Find non-zero values of $x$ satisfying the matrix equation $x\left[ {\begin{array}{cccccccccccccccccccc}{2x}&2\\3&x\end{array}} \right] + 2\left[ {\begin{array}{cccccccccccccccccccc}8&{5x}\\4&{4x}\end{array}} \right] = 2\left[ {\begin{array}{cccccccccccccccccccc}{\left( {{x^2} + 8} \right)}&{24}\\{(10)}&{6x}\end{array}} \right]$.

Given that,
$x\left[ {\begin{array}{cccccccccccccccccccc}{2x}&2\\3&x\end{array}} \right] + 2\left[ {\begin{array}{llllllllllllllllllll}8&{5x}\\4&{4x}\end{array}} \right] = 2\left[ {\begin{array}{cccccccccccccccccccc}{\left( {{x^2} + 8} \right)}&{24}\\{10}&{6x}\end{array}} \right]$

$\Rightarrow$ $\left[ {\begin{array}{cccccccccccccccccccc}{2{x^2}}&{2x}\\{3x}&{{x^2}}\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}{16}&{10x}\\8&{8x}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{2{x^2} + 16}&{48}\\{20}&{12x}\end{array}} \right]$

$\Rightarrow$ $\left[ {\begin{array}{cccccccccccccccccccc}{2{x^2} + 16}&{2x + 10x}\\{3x + 8}&{{x^2} + 8x}\end{array}} \right] = \left[ {\begin{array}{cccccccccccccccccccc}{2{x^2} + 16}&{48}\\{20}&{12x}\end{array}} \right]$

$\Rightarrow$ $2x + 10x = 48$
$\Rightarrow$ $12x = 48$
$\therefore$ $x = \frac{{48}}{{12}} = 4$