Chapter 11, Problem 16CT

To determine

The solution of the system of linear equations using matrices,

$\{\begin{array}{l}2x+2y-3z=5\\ x-y+2z=8\\ 3x+5y-8z=-2\end{array}$.

Answer to Problem 16CT

Solution:

The system of equation $\{\begin{array}{l}2x+2y-3z=5\\ x-y+2z=8\\ 3x+5y-8z=-2\end{array}$ has no solution and therefore is inconsistent.

Explanation of Solution

Given information:

The system of linear equations, $\{\begin{array}{l}2x+2y-3z=5\\ x-y+2z=8\\ 3x+5y-8z=-2\end{array}$.

Explanation:

The augmented matrix of the system is,

$\left[\begin{array}{cccc}2& 2& -3& 5\\ 1& -1& 2& 8\\ 3& 5& -8& -2\end{array}\right]$.

Perform the row operation $R_1=r_2,R_2=r_1$,

$\Rightarrow \left[\begin{array}{cccc}1& -1& 2& 8\\ 2& 2& -3& 5\\ 3& 5& -8& -2\end{array}\right]$

Perform the row operation $R_2=-2r_1+r_2,R_3=-3r_1+r_3$,

$\Rightarrow \left[\begin{array}{cccc}1& -1& 2& 8\\ 0& 4& -7& -11\\ 0& 8& -14& -26\end{array}\right]$

Perform the row operation $R_2=\frac{1}{4}{r}_2$,

$\Rightarrow \left[\begin{array}{cccc}1& -1& 2& 8\\ 0& 1& -\frac{7}{4}& -\frac{11}{4}\\ 0& 8& -14& -26\end{array}\right]$

Perform the row operation $R_3=-8r_2+r_3$,

$\Rightarrow \left[\begin{array}{cccc}1& -1& 2& 8\\ 0& 1& -\frac{7}{4}& -\frac{11}{4}\\ 0& 0& 0& -4\end{array}\right]$

The matrix is in row echelon form the last rowrepresents equation $0=-4$ but it is not possible.

Therefore, the system has no solution and system is inconsistent.

Hence, the system of equation $\{\begin{array}{l}2x+2y-3z=5\\ x-y+2z=8\\ 3x+5y-8z=-2\end{array}$ has no solution and therefore is inconsistent.