Chapter 11.2, Problem 31AYU

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.

$\left[\begin{array}{c}1\\ 0\\ 0\end{array}\begin{array}{c}0\\ 1\\ 0\end{array}\begin{array}{c}0\\ 0\\ 1\end{array}\begin{array}{c}0\\ 1\\ 2\end{array}|\begin{array}{c}1\\ 2\\ 3\end{array}\right]$

To determine

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

$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\begin{array}{c}0\\ 1\\ 2\end{array}|\begin{array}{c}1\\ 2\\ 3\end{array}\right)$

Answer to Problem 31AYU

Solution:

$x+y+w=3$

$y+z+3w=5$

$x+z+2w=4$

Consistent system of equations, infinitely many solutions.

Explanation of Solution

Given:

$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\begin{array}{c}0\\ 1\\ 2\end{array}|\begin{array}{c}1\\ 2\\ 3\end{array}\right)$

Calculation:

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

$x+y+w=3$

$y+z+3w=5$

$x+z+2w=4$

$(R_1=r_1+r_2;R_2=r_2+r_3;R_3=r_3+r_1)$

We find that the rank of the coefficient matrix is 3.

The rank of augmented matrix is also 3.

Both the ranks are equal and is equal to 3.

Here we find that the number of variables is 4 and the number of equations is 3.

So number of variables is more than the number of equations.

Hence the system of equations is consistent or has infinitely many solutions.