Chapter 8, Problem 40RE

To determine

Find thevalue of $x$ and $y$ using the equality of matrices.

Answer to Problem 40RE

The value of $x=12\text{and}y=-2$ .

Explanation of Solution

Given information:

The givenmatrix equation is

  $\left[\begin{array}{cccc}-9& 4& 2& -5\\ 0& -3& 7& -4\\ 6& -1& 1& 0\end{array}\right]=\left[\begin{array}{cccc}-9& 4& x-10& -5\\ 0& -3& 7& 2y\\ \frac{1}{2}x& -1& 1& 0\end{array}\right]$

Calculation:

Two matrices $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ are equal when they have the same order $m\times n$ and $a_{ij}=b_{ij}$ for $1\le i\le m$ and $1\le j\le n$ . In other words, two matrices are equal when their corresponding entries are equal.

The given matrix equation is $\left[\begin{array}{cccc}-9& 4& 2& -5\\ 0& -3& 7& -4\\ 6& -1& 1& 0\end{array}\right]=\left[\begin{array}{cccc}-9& 4& x-10& -5\\ 0& -3& 7& 2y\\ \frac{1}{2}x& -1& 1& 0\end{array}\right]$ and both matrices are equal.

Two matrices are equal when their corresponding entries are equal.

Now the system of equations are

  $\begin{array}{l}2=x-10\\ -4=2y\end{array}$

  $6=\frac{1}{2}x$

Solve the equation and find the value.

  $\begin{array}{l}2=x-10\\ \Rightarrow 2+10=x-10+10\\ \Rightarrow 12=x\\ \Rightarrow x=12\end{array}$

  $\begin{array}{l}-4=2y\\ \Rightarrow \frac{-4}{2}=\frac{2}{2}y\\ \Rightarrow -2=y\\ \Rightarrow y=-2\end{array}$

  $\begin{array}{l}6=\frac{1}{2}x\\ \Rightarrow 6\cdot 2=2\cdot \frac{1}{2}x\\ \Rightarrow 12=x\\ \Rightarrow x=12\end{array}$

Hence thevalue of $x=12\text{and}y=-2$ .