Chapter 11.2, Problem 23AYU

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.

$\left[\begin{array}{c}5\\ 2\\ -4\end{array}\begin{array}{c}-3\\ -5\\ 1\end{array}\begin{array}{c}1\\ 6\\ 4\end{array}|\begin{array}{c}-2\\ -2\\ 6\end{array}\right]$ $\begin{array}{l}{R}_1=-2r_2+r_1\\ R_3=2r_2+r_3\end{array}$

To determine

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

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

Answer to Problem 23AYU

Solution:

$5x-3y+z=-2$

$2x-5y+6z=-2$

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

$\left(\begin{array}{ccc}5& -3& 1\\ -8& 1& 4\\ 0& -9& 16\end{array}|\begin{array}{c}-2\\ 2\\ 2\end{array}\right)$

Explanation of Solution

Given:

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

Calculation:

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

$5x-3y+z=-2$

$2x-5y+6z=-2$

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

The augmented matrix after the row operations $R_2=-2r_1+r_2;R_3=2r_2+r_3$is:

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

$=\left(\begin{array}{ccc}5& -3& 1\\ -10+2& 6-5& -2+6\\ 4-4& -10+1& 12+4\end{array}|\begin{array}{c}-2\\ 4-2\\ -4+6\end{array}\right)$

$=\left(\begin{array}{ccc}5& -3& 1\\ -8& 1& 4\\ 0& -9& 16\end{array}|\begin{array}{c}-2\\ 2\\ 2\end{array}\right)$