Chapter 9, Problem 1RE

Write a system of linear equations represented by the augmented matrix. Then write the solution set.

$\left[\begin{array}{rrrr}\hfill 1& \hfill -2& \hfill 3& \hfill -1\\ \hfill 0& \hfill 1& \hfill 4& \hfill -11\\ \hfill 0& \hfill 0& \hfill 1& \hfill -2\end{array}\right]$

To determine

To calculate:The system of linear equations represented by the augmented matrix.

$\left[\begin{array}{cccc}1& -2& 3& -1\\ 0& 1& 4& 11\\ 0& 0& 1& -2\end{array}\right]$

Also, write its solution set.

Answer to Problem 1RE

The system of linear equations represented by the given augmented matrix is:

$\begin{array}{c}x-2y+3z=-1\\ y+4z=-11\\ z=-2\end{array}$

And the solution set of the given augmented matrix is $\left\{\left(-1,-3,-2\right)\right\}$ .

Explanation of Solution

Calculation:

Consider the given augmented matrix $\left[\begin{array}{cccc}1& -2& 3& -1\\ 0& 1& 4& 11\\ 0& 0& 1& -2\end{array}\right]$ .

Now, to construct the system of linear equations represented by the given augmented matrix, the elements of the augmented matrix on the left side of the vertical bar are the coefficients of variable terms in the equation and on the right side of the vertical bar are the constant terms.

Then, the linear equations can be written as:

$\begin{array}{c}x-2y+3z=-1\\ 0x+y+4z=-11\\ 0x+0y+z=-2\end{array}$

Solve further as:

$\begin{array}{c}x-2y+3z=-1\\ y+4z=-11\\ z=-2\end{array}$

Also, solve for the solution set of the given augmented matrix as:

$\left[\begin{array}{cccc}1& -2& 3& -1\\ 0& 1& 4& 11\\ 0& 0& 1& -2\end{array}\right]$

Apply operation $2{\text{R}}_2+{\text{R}}_1\to {\text{R}}_1$ on above matrix as:

$\left[\begin{array}{cccc}1& 0& 11& -23\\ 0& 1& 4& 11\\ 0& 0& 1& -2\end{array}\right]$

Apply operations ${\text{-11R}}_3+{\text{R}}_1\to {\text{R}}_1{\text{ and -4R}}_3+{\text{R}}_2\to {\text{R}}_2$ on above matrix as:

$\left[\begin{array}{cccc}1& 0& 0& -1\\ 0& 1& 0& -3\\ 0& 0& 1& -2\end{array}\right]$

So, the solution set is $\left\{\left(-1,-3,-2\right)\right\}$ .

Therefore, the system of linear equations represented by the given augmented matrix is:

$\begin{array}{c}x-2y+3z=-1\\ y+4z=-11\\ z=-2\end{array}$

And the solution set of the given augmented matrix is $\left\{\left(-1,-3,-2\right)\right\}$ .