Chapter 2, Problem 16RP

Explanation of Solution

Determining the number of solutions for the given system of equations:

Consider the following linear system,

$\left[\begin{array}{l}\text{1 1 0 0}\\ \text{0 0 1 1}\\ \text{1 0 1 0}\\ \text{0 1 0 1}\end{array}\right]\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{l}2\\ 3\\ 4\\ 1\end{array}\right]$

 We know, $Ax=B$.

The augmented matrix is as given below:

$A|b=\left(\begin{array}{ccccc}1& 1& 0& 0& 2\\ 0& 0& 1& 1& 3\\ 1& 0& 1& 0& 4\\ 0& 1& 0& 1& 1\end{array}\right)$

Now, apply Gauss – Jordan method.

Interchange row 2 and row 4 of $A|b$, we get

$A_1|b_1=\left(\begin{array}{ccccc}1& 1& 0& 0& 2\\ 0& 1& 0& 1& 1\\ 1& 0& 1& 0& 4\\ 0& 0& 1& 1& 3\end{array}\right)$

Replace row 3 by (row 3 – row 1) of $A_1|b_1$, we get

$A_2|b_2=(\begin{array}{ccccc}1& 1& 0& 0& \end{array}$