Question

If we have the summation of matrices $2\left[ \begin{matrix}
   3 & 4 \\
   5 & x \\
\end{matrix} \right]+\left[ \begin{matrix}
   1 & y \\
   0 & 1 \\
\end{matrix} \right]=\left[ \begin{matrix}
   7 & 0 \\
   10 & 5 \\
\end{matrix} \right]$ find the values of x and y.

Answer 1

Hint: For solving this question you should know about the matrices and the addition of matrices. In this problem we will take the addition of LHS of the matrices as a single matrix and then we will compare that LHS with the RHS and find the values of x and y at same place in RHS matrix.

Complete step-by-step solution:
According to our question it is asked of us if, $2\left[ \begin{matrix}
   3 & 4 \\
   5 & x \\
\end{matrix} \right]+\left[ \begin{matrix}
   1 & y \\
   0 & 1 \\
\end{matrix} \right]=\left[ \begin{matrix}
   7 & 0 \\
   10 & 5 \\
\end{matrix} \right]$ , then find the values of x and y. So, as we know that the question is given in a form of matrices and if we want to calculate the value of x and y, then we have to solve this as a single -single matrix form on both sides. So, if we add both the matrices to each other and before that we will do all the calculations of multiplication and division in that and then we add to the solved matrices and will get a final matrix in LHS. And also s single matrix is available in RHS. So, if we compare both the matrices with each other, then these both will provide the answer. For that, we will write it as:
$\begin{align}
  & \Rightarrow 2\left[ \begin{matrix}
   3 & 4 \\
   5 & x \\
\end{matrix} \right]+\left[ \begin{matrix}
   1 & y \\
   0 & 1 \\
\end{matrix} \right]=\left[ \begin{matrix}
   6 & 8 \\
   10 & 2x \\
\end{matrix} \right]+\left[ \begin{matrix}
   1 & y \\
   0 & 1 \\
\end{matrix} \right] \\
 & \Rightarrow \left[ \begin{matrix}
   6+1 & 8+y \\
   10+0 & 2x+1 \\
\end{matrix} \right]=\left[ \begin{matrix}
   7 & 8+y \\
   10 & 2x+1 \\
\end{matrix} \right] \\
 & \Rightarrow \left[ \begin{matrix}
   7 & 8+y \\
   10 & 2x+1 \\
\end{matrix} \right]=\left[ \begin{matrix}
   7 & 0 \\
   10 & 5 \\
\end{matrix} \right] \\
\end{align}$
Now if we compare the left hand side and right hand side with each other, then:
$\begin{align}
  & 8+y=0 \\
 &\Rightarrow y=-8 \\
 & 2x+1=5 \\
 &\Rightarrow 2x=4 \\
 &\Rightarrow x=2 \\
\end{align}$
So, the values of y = -8 and x = 2.

Note: For solving these type of questions you should be careful of the calculations of matrices because if you add to nay matrices with each other and if anyone is multiplied by some digit, then that will be multiplied by new whole number which will make our question wrong. And if we multiply that before and then we add, then our answer will be completely right and many minor mistakes like this are to be kept in mind while solving this type of question.