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

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,

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.