Chapter 2.3, Problem 23E

To determine

To find: value of $x\text{ and }y$ for the given matrix equation.

Answer to Problem 23E

  $\begin{array}{l}x=-1\\ y=1\end{array}$

Explanation of Solution

Given information:

The given matrix equation is as follows:

  $\left[\begin{array}{cc}\begin{array}{l}x+y\\ y\end{array}& \begin{array}{l}3\\ 6\end{array}\end{array}\right]=\left[\begin{array}{cc}0& 2y-x\\ y^2& 4-2x\end{array}\right]$

Calculation:

Since the corresponding elements are equal, so the equality of the matrices can be represented as follows:

  $\begin{array}{c}x+y=0\mathrm{...}\left(1\right)\\ y=y^2\mathrm{...}\left(2\right)\\ 3=2y-x\mathrm{...}\left(3\right)\\ 6=4-2x\mathrm{...}\left(4\right)\end{array}$

Solve equation (2) as follows:

  $\begin{array}{c}6=4-2x\\ 2x=4-6\\ 2x=-2\\ x=\frac{-2}{2}\\ =-1\end{array}$

Put value of x in equation (1) and solve for y as follows:

  $\begin{array}{c}-1+y=0\\ y=1\end{array}$

Thus, the values of x and y are $-1$ and $1$ respectively.