Chapter 11.2, Problem 20AYU

In Problems 19-26, write the system of equations corresponding to each augmented matrix. Then perform the indicated row operation(s) on the given augmented matrix.

$\left[\begin{array}{c}1\\ -4\\ -3\end{array}\begin{array}{c}-3\\ -5\\ -2\end{array}\begin{array}{c}3\\ -3\\ 4\end{array}|\begin{array}{c}-5\\ -5\\ 6\end{array}\right]$ $\begin{array}{l}{R}_2=4r_1+r_2\\ R_3=3r_1+r_3\end{array}$

To determine

To find: The system of equations corresponding to the given augmented matrix and perform the indicated row operation on this augmented matrix:

$\left(\begin{array}{ccc}1& -3& 3\\ -4& -5& -3\\ -3& -2& 4\end{array}|\begin{array}{c}-5\\ -5\\ 6\end{array}\right);R_2=4r_1+r_2;R_3=3r_1+r_3$

Answer to Problem 20AYU

Solution:

$x-3y+3z=-5$

$-4x-5y-3z=-5$

$-3x-2y+4z=6$

$\left(\begin{array}{ccc}1& -3& 3\\ 0& -17& 9\\ 0& -11& 13\end{array}|\begin{array}{c}-5\\ -25\\ -9\end{array}\right)$

Explanation of Solution

Given:

$\left(\begin{array}{ccc}1& -3& 3\\ -4& -5& -3\\ -3& -2& 4\end{array}|\begin{array}{c}-5\\ -5\\ 6\end{array}\right);R_2=4r_1+r_2;R_3=3r_1+r_3$

Calculation:

The system of equations corresponding to the given augmented matrix is:

$x-3y+3z=-5$

$-4x-5y-3z=-5$

$-3x-2y+4z=6$

The augmented matrix after the row operations $R_2=4r_1+r_2;R_3=3r_1+r_3$ is:

$\left(\begin{array}{ccc}1& -3& 3\\ 4\left(1\right)-4& 4\left(-3\right)-5& 4\left(3\right)-3\\ 3\left(1\right)-3& 3\left(-3\right)-2& 3\left(3\right)+4\end{array}|\begin{array}{c}-5\\ 4\left(-5\right)-5\\ 3\left(-5\right)+6\end{array}\right)$

$=\left(\begin{array}{ccc}1& -3& 3\\ 4-4& -12-5& 12-3\\ 3-3& -9-2& 9+4\end{array}|\begin{array}{c}-5\\ -20-5\\ -15+6\end{array}\right)$

$=\left(\begin{array}{ccc}1& -3& 3\\ 0& -17& 9\\ 0& -11& 13\end{array}|\begin{array}{c}-5\\ -25\\ -9\end{array}\right)$