Chapter 11.2, Problem 35AYU

In Problems 27-38, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use $x,y$; or $x,y,z$; or $x_1,x_2,x_3,x_4$ as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.

$\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\begin{array}{c}1\\ 2\\ -1\\ 0\end{array}|\begin{array}{c}-2\\ 2\\ 0\\ 0\end{array}\right]$

To determine

To find: The system of equations corresponding to the given reduced row echelon augmented matrix and find the solution, if possible:

$\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 2\\ 0& 0& 1& -1\\ 0& 0& 0& 0\end{array}|\begin{array}{c}\begin{array}{c}-2\\ 2\end{array}\\ \begin{array}{c}0\\ 0\end{array}\end{array}\right)$

Answer to Problem 35AYU

Solution:

$x+w=-2$

$y+2w=2$

$z-w=0$

Consistent system of equations, infinitely many solutions.

Explanation of Solution

Given:

$\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& 1& 0& 2\\ 0& 0& 1& -1\\ 0& 0& 0& 0\end{array}|\begin{array}{c}\begin{array}{c}-2\\ 2\end{array}\\ \begin{array}{c}0\\ 0\end{array}\end{array}\right)$

Calculation:

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

$x+w=-2$

$y+2w=2$

$z-w=0$

We find that the rank of the coefficient matrix is 3, less than 4.

The rank of augmented matrix is also 3, less than 4.

And both the ranks are equal.

Hence the above system is consistent and has infinitely many solutions.