Chapter 7.5, Problem 12E

To determine

To find: the values of $x,y,\text{and}z$ .

Answer to Problem 12E

The solution is $x=-5,y=-4,z=9$ .

Explanation of Solution

Given information:

A matrix equation is given as

  $\left[\begin{array}{ccc}x+4& 8& -3\\ 1& 22& 2y\\ 7& -2& z+2\end{array}\right]=\left[\begin{array}{ccc}2x+9& 8& -3\\ 1& 22& -8\\ 7& -2& 11\end{array}\right]$

Concept used:

If two matrices of equal dimension are equal then the corresponding elements of the matrices are also equal.

Dimension of matrix with a rows and b columns is given by $a\times b$ .

Two matrices $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ are equal when they have same dimension and their corresponding entries are equal.

Calculation:

Consider the given matrix equation.

  $\left[\begin{array}{ccc}x+4& 8& -3\\ 1& 22& 2y\\ 7& -2& z+2\end{array}\right]=\left[\begin{array}{ccc}2x+9& 8& -3\\ 1& 22& -8\\ 7& -2& 11\end{array}\right]$

Now, corresponding elements of the above two matrices are equal.

  $\begin{array}{l}x+4=2x+9\mathrm{......}(1)\\ 2y=-8\mathrm{......}(2)\\ z+2=11\mathrm{......}(3)\end{array}$

From equation (1),

  $\begin{array}{c}x+4=2x+9\\ 2x-x=4-9\\ x=-5\end{array}$

From equation (2),

  $\begin{array}{l}2y=-8\\ y=\frac{-8}{2}\\ y=-4\end{array}$

From equation (3),

  $\begin{array}{c}z+2=11\\ z=11-2\\ z=9\end{array}$

So, the unknown values can be obtained as follows:

  $\begin{array}{l}x=-5\\ y=-4\\ z=9\end{array}$

Therefore, the values of $x,y\text{and}z$ is

  $x=-5,y=-4,z=9$