Question

If $\left( \begin{matrix}
   x & 0 \\
   1 & y \\
\end{matrix} \right)+\left( \begin{matrix}
   -2 & 1 \\
   3 & 4 \\
\end{matrix} \right)=\left( \begin{matrix}
   3 & 5 \\
   6 & 3 \\
\end{matrix} \right)-\left( \begin{matrix}
   2 & 4 \\
   2 & 1 \\
\end{matrix} \right)$ then
A . $x=-3,y=-2$
B. $x=3,y=-2$
C. $x=3,y=2$
D. $x=-3,y=2$

Answer 1

Hint:In this question, we have given the matrices, and all they are of $2\times 2$ order and we have to find the value of x and y. to find the values, we have to add and subtract the matrices. First, we solve the left-hand side by adding the matrices, then we solve the right-hand side by subtracting the matrices. Then we equate both equations and simplify them to get the value of x and y.

Complete step by step Solution:
Given $\left( \begin{matrix}
   x & 0 \\
   1 & y \\
\end{matrix} \right)+\left( \begin{matrix}
   -2 & 1 \\
   3 & 4 \\
\end{matrix} \right)=\left( \begin{matrix}
   3 & 5 \\
   6 & 3 \\
\end{matrix} \right)-\left( \begin{matrix}
   2 & 4 \\
   2 & 1 \\
\end{matrix} \right)$
All the matrices are of $2\times 2$ order.
We have to find the value of x and y.
To find the value of x and y, we can add and subtract the given matrices.
First we add the matrices $\left( \begin{matrix}
   x & 0 \\
   1 & y \\
\end{matrix} \right)$ and $\left( \begin{matrix}
   -2 & 1 \\
   3 & 4 \\
\end{matrix} \right)$
Then $\left( \begin{matrix}
   x & 0 \\
   1 & y \\
\end{matrix} \right)$ + $\left( \begin{matrix}
   -2 & 1 \\
   3 & 4 \\
\end{matrix} \right)$ = $\left( \begin{matrix}
   x+(-2) & 0+1 \\
   1+3 & y+4 \\
\end{matrix} \right)$
Hence $\left( \begin{matrix}
   x & 0 \\
   1 & y \\
\end{matrix} \right)$ + $\left( \begin{matrix}
   -2 & 1 \\
   3 & 4 \\
\end{matrix} \right)$ = $\left( \begin{matrix}
   x-2 & 1 \\
   4 & y+4 \\
\end{matrix} \right)$
Now we subtract the matrices $\left( \begin{matrix}
   3 & 5 \\
   6 & 3 \\
\end{matrix} \right)$ and $\left( \begin{matrix}
   2 & 4 \\
   2 & 1 \\
\end{matrix} \right)$
$\left( \begin{matrix}
   3 & 5 \\
   6 & 3 \\
\end{matrix} \right)$ - $\left( \begin{matrix}
   2 & 4 \\
   2 & 1 \\
\end{matrix} \right)$ = $\left( \begin{matrix}
   3-2 & 5-4 \\
   6-2 & 3-1 \\
\end{matrix} \right)$
$\left( \begin{matrix}
   3 & 5 \\
   6 & 3 \\
\end{matrix} \right)$ - $\left( \begin{matrix}
   2 & 4 \\
   2 & 1 \\
\end{matrix} \right)$ = $\left( \begin{matrix}
   1 & 1 \\
   4 & 2 \\
\end{matrix} \right)$
Now we equate both the matrices and get
$\left( \begin{matrix}
   x-2 & 1 \\
   4 & y+4 \\
\end{matrix} \right)$ = $\left( \begin{matrix}
   1 & 1 \\
   4 & 2 \\
\end{matrix} \right)$
Now we compute and simplify the above terms, and we get
$x-2=1$ and
$y+4=2$
By simplifying the above equations, we get
$x=3,y=-2$
Hence the value of $x=3,y=-2$

Therefore, the correct option is (B).

Note: Keep in mind before adding and subtracting any matrices that they have an equal number of columns and rows to be added. As the given matrices are of order $2\times 2$, so we can add it simply. Similarly we can add a $2\times 3$ matrix with a $2\times 3$ matrix or $3\times 3$ matrix with $3\times 3$ matrix. However, we cannot add $2\times 3$ matrix with a $3\times 2$ matrix. Similarly, we cannot add $2\times 2$ matrix with a $3\times 3$ matrix. The order in which we add the matrix is not important because the addition of two matrices is commutative.