Chapter 11.2, Problem 52AYU

To determine

To find: The solution to the given system of equations using matrices.

Answer to Problem 52AYU

Solution:

No solutions, Inconsistent system.

Explanation of Solution

Given:

$2x-3y-z=0$

$-x+2y+z=5$

$3x-4y-z=1$

Formula used:

To solve a system of three equations in $x,y$ and $z$ using matrices:

Step 1: Write the corresponding matrix associated with the system of equations.

Step 2: Use elementary row operations to get equivalent matrix of the form:

$\left[\begin{array}{ccc}1& a& b\\ 0& 1& c\\ 0& 0& 1\end{array}|\begin{array}{c}d\\ e\\ f\end{array}\right]$; where $a,b,c,d,e,f$ are constants.

Step 3: Solve for $x,y$ and $z$.

Calculation:

$2x-3y-z=0$

$-x+2y+z=5$

$3x-4y-z=1$

The corresponding augmented matrix is:

$\left[\begin{array}{ccc}2& -3& -1\\ -1& 2& 1\\ 3& -4& -1\end{array}|\begin{array}{c}0\\ 5\\ 1\end{array}\right]$

$\approx \left[\begin{array}{ccc}1& -\frac{3}{2}& -\frac{1}{2}\\ -1& 2& 1\\ 3& -4& -1\end{array}|\begin{array}{c}0\\ 5\\ 1\end{array}\right];R_1=\frac{1}{2}{r}_1$

$\approx \left[\begin{array}{ccc}1& -\frac{3}{2}& -\frac{1}{2}\\ 0& \frac{1}{2}& \frac{1}{2}\\ 3& -4& -1\end{array}|\begin{array}{c}0\\ 5\\ 1\end{array}\right];R_2=R_1+r_2$

$\approx \left[\begin{array}{ccc}1& -\frac{3}{2}& -\frac{1}{2}\\ 0& \frac{1}{2}& \frac{1}{2}\\ 0& \frac{1}{2}& \frac{1}{2}\end{array}|\begin{array}{c}0\\ 5\\ 1\end{array}\right];R_3=-3R_1+r_3$

$\approx \left[\begin{array}{ccc}1& -\frac{3}{2}& -\frac{1}{2}\\ 0& 1& 1\\ 0& 1& 1\end{array}|\begin{array}{c}0\\ 10\\ 2\end{array}\right];R_2^{\text{'}}=2R_2;R_3^{\text{'}}=2R_3$

$\approx \left[\begin{array}{ccc}1& -\frac{3}{2}& -\frac{1}{2}\\ 0& 1& 1\\ 0& 0& 0\end{array}|\begin{array}{c}0\\ 10\\ -8\end{array}\right];R_3^{\text{'}\text{'}}=R_2^{\text{'}}+R_3\text{'}$

$\approx \left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 0& 0& 0\end{array}|\begin{array}{c}30\\ 10\\ -8\end{array}\right];R_1^{\text{'}}=\frac{3}{2}{R}_2^{\text{'}}+R_1$

$\approx \left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 0& 0& 0\end{array}|\begin{array}{c}30\\ 10\\ 1\end{array}\right];R_3^{\text{'}\text{'}\text{'}}=-\frac{1}{8}{R}_3\text{'}\text{'}$

Thus we see that the rank of the coefficient matrix is 2, but the rank of augmented matrix is 3.

Since rank of the coefficient matrix is less than the rank of the augmented matrix, therefore the given system of equations has no solutions.

Hence this is an inconsistent system.