Question

If two matrices are equal as \[\left( \begin{matrix}
   x-y & z \\
   2x-y & w \\
\end{matrix} \right)=\left( \begin{matrix}
   -1 & 4 \\
   0 & 5 \\
\end{matrix} \right)\], Then find \[x,y,z,w\]

Answer 1

Hint: It is given that two matrices are equal. If two matrices are equal then their principle elements are the same. So we have equated the principal element of the first matrix to the second matrix. Solving the equations gets the values of \[x,y,z,w\].

Complete step-by-step solution -
Given matrix is \[\left( \begin{matrix}
   x-y & z \\
   2x-y & w \\
\end{matrix} \right)=\left( \begin{matrix}
   -1 & 4 \\
   0 & 5 \\
\end{matrix} \right)\].
Now equating the principal element of first matrix to second matrix, we get the equations as:
\[x-y=-1\]. . . . . . . . . . . . . . . . . . . . . (1)
\[z=4\]
\[2x-y=0\]. . . . . . . . . . . . . . . . . . . . . . . (2)
\[w=5\]
The values of w and z are known, so from the equations (1) and (2) we have to solve for x and y.
Writing both the equations of (1) and (2), equations are shown below:
\[x-y=-1\]
\[2x-y=0\]
Subtracting equation (2) from equation (1) we get the solution as follows:
\[x-y-(2x-y)=-1-0\]
By further solving, we get
\[x-2x=-1\]
\[-x=-1\]
\[x=1\]
We have got the value of x. Now by substituting the value of x in the equation (1) we get the value of y as
\[1-y=-1\]
\[y=1+1\]
\[y=2\]
The value of x and y from equation (1) and (2) are \[x=1\],\[y=2\].

Note: This is a direct problem in matrices. The two given matrices are equated, the first element in the first matrix is equal to the first element in the second matrix. By solving the equation we get the values. Equating process should be in order. There is a need to remember that two matrices are to be equal when they have the same order and same elements.