Chapter 2, Problem 1RP

Explanation of Solution

Using Gauss-Jordan method to indicate the solutions:

Consider the given system of linear equations,

$\begin{array}{l}{x}_1+x_2=2\\ x_2+x_3=3\\ x_1+2x_2+x_3=5\end{array}$

The augmented matrix of this system is as follows:

$A|b=\left[\begin{array}{l}110\\ 011\\ 121\end{array}\right]\left[\begin{array}{l}2\\ 3\\ 5\end{array}\right]$

The Gauss-Jordan method is applied to find the solutions of the above system of linear equations.

Replace row 3 of A|b by (row 3 – row 1), then the following matrix is obtained,

$A_1|b_1=\left[\begin{array}{l}110\\ 011\\ 011\end{array}\right]\left[\begin{array}{l}2\\ 3\\ 3\end{array}\right]$

Now, replace row 3 of $A_1|b_1$ by (row 3 - row 2), then the following matrix is obtained,

$A_2|b_2=\left[\begin{array}{l}110\\ 011\\ 000\end{array}\right]\left[\begin{array}{l}2\\ 3\\ 0\end{array}\right]$

This produces the following result,

$\begin{array}{l}{x}_1+x_2=2\\ x_2+x_3=3\end{array}$

Solving the above equations, the following result is obtained,

$\begin{array}{l}{x}_1=k-1\\ x_2=3-k\\ x_3=k\end{array}$

Therefore, the above system of linear equations has a unique solution.