Question

Find the value of x + y if \[\left[ \begin{matrix}
   x-y & z \\
   2x-y & w \\
\end{matrix} \right]=\left[ \begin{matrix}
   -1 & 4 \\
   0 & 5 \\
\end{matrix} \right].\]

Answer 1

Hint: We are given two matrices \[\left[ \begin{matrix}
   x-y & z \\
   2x-y & w \\
\end{matrix} \right]\And \left[ \begin{matrix}
   -1 & 4 \\
   0 & 5 \\
\end{matrix} \right]\] that are equal. Since matrices are equal, so their corresponding entries/elements must be equal. So, we will compare both the matrices and we will get, x – y = – 1 and 2x – y = 0. Now using the substitution method, 2x = y, we will solve for x and then for y and then find the value for x + y.

Complete step by step answer:
We are given two matrices \[\left[ \begin{matrix}
   x-y & z \\
   2x-y & w \\
\end{matrix} \right]\And \left[ \begin{matrix}
   -1 & 4 \\
   0 & 5 \\
\end{matrix} \right]\] and both the matrices are given as equal and we have to find the value of x + y. We know that two matrices are said to be equal if the corresponding elements of both the matrices are equal.
As we are given that matrix \[\left[ \begin{matrix}
   x-y & z \\
   2x-y & w \\
\end{matrix} \right]=\left[ \begin{matrix}
   -1 & 4 \\
   0 & 5 \\
\end{matrix} \right]\] so means that their corresponding entries/elements are the same. Now, we will compare both the matrices to find the required values. We have,
\[\left[ \begin{matrix}
   x-y & z \\
   2x-y & w \\
\end{matrix} \right]=\left[ \begin{matrix}
   -1 & 4 \\
   0 & 5 \\
\end{matrix} \right]\]
Comparing both the matrix, we get,
\[x-y=-1.....\left( i \right)\]
\[2x-y=0.....\left( ii \right)\]
\[z=4\]
\[w=5\]
From (ii), we will get,
\[\Rightarrow 2x=y\]
Now, we will use this 2x = y to solve further. Putting y = 2x in (i), we will get,
\[\Rightarrow x-2x=-1\]
\[\Rightarrow -x=-1\]
Cancelling the negative on both the sides, we will get,
\[\Rightarrow x=1\]
Now, as x = 1, putting it in y = 2x, we will get,
\[\Rightarrow y=2\times 1\]
\[\Rightarrow y=2\]

So, as x = 1 and y = 2, we will get,
\[\Rightarrow x+y=1+2=3\]

Note: While solving the matrix, always remember that \[\left| \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right|\] is a sign for the determinant of the matrix while \[\left[ \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right]\] is a symbol. In our question, we are given that the matrices are equal, so we just need to compare the term, and finding the determinant is not required.