# Find the value of $$x$$ and $$y$$ if : $2\begin{bmatrix} 1 & 3 \\[0.3em] 0 & x \\[0.3em] \end{bmatrix} + \begin{bmatrix} y & 0 \\[0.3em] 1 & 2 \\[0.3em] \end{bmatrix} = \begin{bmatrix} 5 & 6 \\[0.3em] 1 & 8 \\[0.3em] \end{bmatrix}$

$\begin{array}{1 1} x = 3 \;y = 2 \\ x = 2\; y = 3 \\ x = 3 \;y = 3 \\ x = 2 \;y = 0 \end{array}$

Toolbox:
• 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.
• We can then match the corresponding elements and solve the resulting equations to find the values of the unknown variables.
• 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.
Step1:
Given:
$2\begin{bmatrix}1 & 3\\0 & x\end{bmatrix}+\begin{bmatrix}y & 0\\1 & 2\end{bmatrix}=\begin{bmatrix}5 & 6\\1 & 8\end{bmatrix}$
$\Rightarrow \begin{bmatrix}2 & 6\\0 & 2x\end{bmatrix}+\begin{bmatrix}y & 0\\1 & 2\end{bmatrix}=\begin{bmatrix}5 & 6\\1 & 8\end{bmatrix}$
$\Rightarrow \begin{bmatrix}2+y & 6+0 \\0+1 & 2x+2\end{bmatrix}=\begin{bmatrix}5 &6\\1 & 8\end{bmatrix}$
Step2:
The given two matrices are equal,hence their corresponding elements should be equal.
$\Rightarrow 2+y=5$
y=5-2
y=3
2x+2=8
2x=8-2
2x=6
x=3