Question

Find x, y satisfying the following matrix equation:
\[\left[ \begin{matrix}
   x & y+2 & z-3 \\
\end{matrix} \right]+\left[ \begin{matrix}
   y & 4 & 5 \\
\end{matrix} \right]=\left[ \begin{matrix}
   4 & 9 & 12 \\
\end{matrix} \right]\].

Answer 1

Hint: We are given a matrix based equation which we have to solve for x and y. We will use the properties of matrices to solve for the given variables. We will add up the terms of the matrices in the LHS and equate it to the constants in the RHS. We will have three equations, which are, \[x+y=4\], \[y+6=9\] and \[z+2=12\]. We will solve these equations and hence, we will have the values of x and y.

Complete step-by-step answer:
According to the given question, we are given an expression based on matrix. We are given an equation using matrix having the variables x, y and z and we are asked to find the values of x and y.
We will be using the properties of the matrix to solve the given equation.
The given equation that we have is,
\[\left[ \begin{matrix}
   x & y+2 & z-3 \\
\end{matrix} \right]+\left[ \begin{matrix}
   y & 4 & 5 \\
\end{matrix} \right]=\left[ \begin{matrix}
   4 & 9 & 12 \\
\end{matrix} \right]\]
In the LHS, we will add the two matrices such that the terms on the similar positions are added. We get,
\[\Rightarrow \left[ \begin{matrix}
   x+y & y+2+4 & z-3+5 \\
\end{matrix} \right]=\left[ \begin{matrix}
   4 & 9 & 12 \\
\end{matrix} \right]\]
Adding up the terms we get,
\[\Rightarrow \left[ \begin{matrix}
   x+y & y+6 & z+2 \\
\end{matrix} \right]=\left[ \begin{matrix}
   4 & 9 & 12 \\
\end{matrix} \right]\]
Now, we will equate the terms of the matrix in LHS to the terms of matrix in RHS, we get the equations as,
\[x+y=4\]-(1)
\[y+6=9\]-(2)
\[z+2=12\]-(3)
From equation (2), we get the value of y as,
\[\Rightarrow y+6-6=9-6\]
\[\Rightarrow y=3\]
We will now substitute the value of \[y=3\] in the equation (1), we get,
\[\Rightarrow x+3=4\]
\[\Rightarrow x+3-3=4-3\]
\[\Rightarrow x=1\]
And from equation (3), we get,
\[\Rightarrow z+2-2=12-2\]
\[\Rightarrow z=10\]
Therefore, the value of \[x=1\], \[y=3\] and \[z=10\].

Note: The matrices when added should not be added haphazardly, that is, the first element of the first matrix is added to the first element of the second matrix and so on. The three equations obtained should be solved step wise and also while substituting the values back in an equation, it should be carried out correctly without values getting changed.