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

Toolbox:
• The sum / difference $A(+/-)B$ of two $m$-by-$n$ matrices $A$ and $B$ is calculated entrywise: $(A (+/-) B)_{i,j} = A_{i,j} +/- B_{i,j}$ where 1 ≤ i ≤ m and 1 ≤ j ≤ n.
• If the order of 2 matrices are equal, their corresponding elements are equal, i.e, if $A_{ij}=B_{ij}$, then any element $a_{ij}$ in matrix A is equal to corresponding element $b_{ij}$ in matrix B.
Step1:
Given
$x\begin{bmatrix}2x & 2\\3 & x\end{bmatrix}+2\begin{bmatrix}8 & 5x\\4 & 4x\end{bmatrix}=2\begin{bmatrix}(x^2+8)& 24\\10 & 6x\end{bmatrix}$
$\begin{bmatrix}2x^2 & 2x\\3x & x^2\end{bmatrix}+\begin{bmatrix}16 & 10x\\8 & 8x\end{bmatrix}=\begin{bmatrix}2(x^2+8)& 48\\20 & 12x\end{bmatrix}$
$\begin{bmatrix}2x^2 & 2x\\3x & x^2\end{bmatrix}+\begin{bmatrix}16 & 10x\\8 & 8x\end{bmatrix}=\begin{bmatrix}2x^2+16& 48\\20 & 12x\end{bmatrix}$
$\begin{bmatrix}2x^2+16 & 2x+10x\\3x+8 & x^2+8x\end{bmatrix}=\begin{bmatrix}2x^2+16 & 48\\20 & 12x\end{bmatrix}$
Step2:
The given two matrices are equal hence the corresponding elements should be equal.
$\Rightarrow 2x^2+16=2x^2+16$
By equating the first element we cannot get the non-zero value as the elements are equal.
Let us equate the $2^{nd}$ element
2x+10x=48.
12x=48
x=48/12
x=4
Step3:
Thus the non zero value of x is x=4.