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

$\begin{array}{1 1} x=4y=4 \\ x=3y =3 \\ x=4y =3 \\ x=3y = 0 \end{array}$

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## 1 Answer

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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.
• We can then match the corresponding elements and solve the resulting equations to find the values of the unknown variables.
Step1:
Given:$2\begin{bmatrix}1 &3\\0 & x\end{bmatrix}+\begin{bmatrix}y & 0\\1 &2\end{bmatrix}=\begin{bmatrix}5 & 6\\1 & 8\end{bmatrix}$
$\Rightarrow \begin{bmatrix}2 &6\\0 & 2x\end{bmatrix}+\begin{bmatrix}y & 0\\1 &2\end{bmatrix}=\begin{bmatrix}5 & 6\\1 & 8\end{bmatrix}$
$\Rightarrow \begin{bmatrix}2+y &0+6\\0+1 & 2x+2\end{bmatrix}=\begin{bmatrix}5 & 6\\1 & 8\end{bmatrix}$
Step2:
Since the given two matrices are equal hence their corresponding elements should be equal.
2+y=5-----(1)
2x+2=8------(2)
From equation (1) we have
2+y=5
y=5-2
y=3
From equation(2) we have
2x+2=8.
2x=8-2
2x=6
x=3.
answered Apr 4, 2013
edited Jul 13

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