Chapter 3.4, Problem 29HP

To determine

To solve a system of four equations with 4 variables.

Explanation of Solution

Given:

System of four equations with 4 variables

Concept used:

To find inverse of a square matrix A , following steps should be followed.

1. Augment given matrix with the identity matrix
2. Perform row operations to make the identity matrix to the left and the right matrix becomes inverse matrix.

In this way, inverse matrix of a matrix can be obtained.

Formula used:

  $AX=B\iff X=A^{-1}B$

Calculation:

Let $x_1,x_2,x_3\text{and}{x}_4$ be the 4 variables.

Consider a system of equations which has 4 variables,

  $\{\begin{array}{l}{x}_1-2x_2-x_3-2x_4=0\\ 3x_1-5x_2-2x_3-3x_4=1\\ 2x_1-5x_2-2x_3-5x_4=-1\\ -x_1+4x_2+4x_3+11x_4=2\end{array}$

Writing above system of linear equations in matrix form $(AX=B)$ ,

  $\left[\begin{array}{cccc}1& -2& -1& 2\\ 3& -5& -2& -3\\ 2& -5& -2& -5\\ -1& 4& 4& 11\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ -1\\ 2\end{array}\right]$

Here,

  $\begin{array}{l}A=\left[\begin{array}{cccc}1& -2& -1& 2\\ 3& -5& -2& -3\\ 2& -5& -2& -5\\ -1& 4& 4& 11\end{array}\right]\\ X=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]\\ B=\left[\begin{array}{c}0\\ 1\\ -1\\ 2\end{array}\right]\end{array}$

Now, find the inverse matrix of matrix A .

Step 1: Augmenting matrix A with the identity matrix

  $[\begin{array}{cccc}1& -2& -1& 2\\ 3& -5& -2& -3\\ 2& -5& -2& -5\\ -1& 4& 4& 11\end{array}\begin{array}{c}\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\end{array}]$

Step 2: Performing row operations to make the identity matrix to the left.

  $[\begin{array}{cccc}1& -2& -1& 2\\ 3& -5& -2& -3\\ 2& -5& -2& -5\\ -1& 4& 4& 11\end{array}\begin{array}{c}\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\end{array}]$

  $\text{Performing}{R}_2=R_2-3R_1$

  $[\begin{array}{cccc}1& -2& -1& 2\\ 0& 1& 1& -9\\ 2& -5& -2& -5\\ -1& 4& 4& 11\end{array}\begin{array}{c}\begin{array}{cccc}1& 0& 0& 0\\ -3& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\end{array}]$

  $\text{Performing}{R}_3=R_3-2R_1$

  $[\begin{array}{cccc}1& -2& -1& 2\\ 0& 1& 1& -9\\ 0& -1& 0& -9\\ -1& 4& 4& 11\end{array}\begin{array}{c}\begin{array}{cccc}1& 0& 0& 0\\ -3& 1& 0& 0\\ -2& 0& 1& 0\\ 0& 0& 0& 1\end{array}\end{array}]$

  $\text{Performing}{R}_4=R_4+R_1$

  $[\begin{array}{cccc}1& -2& -1& 2\\ 0& 1& 1& -9\\ 0& -1& 0& -9\\ 0& 2& 3& 13\end{array}\begin{array}{c}\begin{array}{cccc}1& 0& 0& 0\\ -3& 1& 0& 0\\ -2& 0& 1& 0\\ 1& 0& 0& 1\end{array}\end{array}]$

  $\text{Performing}{R}_1=R_1+2R_2$

  $[\begin{array}{cccc}1& 0& 1& -16\\ 0& 1& 1& -9\\ 0& -1& 0& -9\\ 0& 2& 3& 13\end{array}\begin{array}{c}\begin{array}{cccc}-5& 2& 0& 0\\ -3& 1& 0& 0\\ -2& 0& 1& 0\\ 1& 0& 0& 1\end{array}\end{array}]$

  $\text{Performing}{R}_3=R_3+R_2$

  $[\begin{array}{cccc}1& 0& 1& -16\\ 0& 1& 1& -9\\ 0& 0& 1& -18\\ 0& 2& 3& 13\end{array}\begin{array}{c}\begin{array}{cccc}-5& 2& 0& 0\\ -3& 1& 0& 0\\ -5& 1& 1& 0\\ 1& 0& 0& 1\end{array}\end{array}]$

  $\text{Performing}{R}_4=R_4-2R_2\mathbb{S}$

  $\left[\begin{array}{cc}\begin{array}{cccc}1& 0& 1& -16\\ 0& 1& 1& -9\\ 0& 0& 1& -18\\ 0& 0& 1& 31\end{array}& \begin{array}{cccc}-5& 2& 0& 0\\ -3& 1& 0& 0\\ -5& 1& 1& 0\\ 7& -2& 0& 1\end{array}\end{array}\right]$

  $\text{Performing}{R}_1=R_1-R_3$

  $[\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 1& -9\\ 0& 0& 1& -18\\ 0& 0& 1& 31\end{array}\begin{array}{c}\begin{array}{cccc}0& 1& -1& 0\\ -3& 1& 0& 0\\ -5& 1& 1& 0\\ 7& -2& 0& 1\end{array}\end{array}]$

  $\text{Performing}{R}_2=R_2-R_3$

  $[\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 0& 9\\ 0& 0& 1& -18\\ 0& 0& 1& 31\end{array}\begin{array}{c}\begin{array}{cccc}0& 1& -1& 0\\ 2& 0& -1& 0\\ -5& 1& 1& 0\\ 7& -2& 0& 1\end{array}\end{array}]$

  $\text{Performing}{R}_4=R_4-R_3$

  $[\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 0& 9\\ 0& 0& 1& -18\\ 0& 0& 0& 49\end{array}\begin{array}{c}\begin{array}{cccc}0& 1& -1& 0\\ 2& 0& -1& 0\\ -5& 1& 1& 0\\ 12& -3& -1& 1\end{array}\end{array}]$

  $\text{Performing}{R}_4=\frac{R_4}{49}$

  $\left[\begin{array}{cc}\begin{array}{cccc}1& 0& 0& 2\\ 0& 1& 0& 9\\ 0& 0& 1& -18\\ 0& 0& 0& 1\end{array}& \begin{array}{cccc}0& 1& -1& 0\\ 2& 0& -1& 0\\ -5& 1& 1& 0\\ \frac{12}{49}& -\frac{3}{49}& -\frac{1}{49}& \frac{1}{49}\end{array}\end{array}\right]$

  $\text{Performing}{R}_1=R_1-2R_4\mathfrak{M}$

  $\left[\begin{array}{cc}\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 9\\ 0& 0& 1& -18\\ 0& 0& 0& 1\end{array}& \begin{array}{cccc}-\frac{24}{49}& \frac{55}{49}& -\frac{47}{49}& -\frac{2}{49}\\ 2& 0& -1& 0\\ -5& 1& 1& 0\\ \frac{12}{49}& -\frac{3}{49}& -\frac{1}{49}& \frac{1}{49}\end{array}\end{array}\right]$

  $\text{Performing}{R}_2=R_2-9R_4$

  $\left[\begin{array}{cc}\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& -18\\ 0& 0& 0& 1\end{array}& \begin{array}{cccc}-\frac{24}{49}& \frac{55}{49}& -\frac{47}{49}& -\frac{2}{49}\\ -\frac{10}{49}& \frac{27}{49}& -\frac{40}{49}& -\frac{9}{49}\\ -5& 1& 1& 0\\ \frac{12}{49}& -\frac{3}{49}& -\frac{1}{49}& \frac{1}{49}\end{array}\end{array}\right]$

  $\text{Performing}{R}_3=R_3+18R_4$

  $\left[\begin{array}{cc}\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}& \begin{array}{cccc}-\frac{24}{49}& \frac{55}{49}& -\frac{47}{49}& -\frac{2}{49}\\ -\frac{10}{49}& \frac{27}{49}& -\frac{40}{49}& -\frac{9}{49}\\ -\frac{29}{49}& -\frac{5}{49}& \frac{31}{49}& \frac{18}{49}\\ \frac{12}{49}& -\frac{3}{49}& -\frac{1}{49}& \frac{1}{49}\end{array}\end{array}\right]$

So, inverse matrix is,

  $A^{-1}=\left[\begin{array}{cccc}-\frac{24}{49}& \frac{55}{49}& -\frac{47}{49}& -\frac{2}{49}\\ -\frac{10}{49}& \frac{27}{49}& -\frac{40}{49}& -\frac{9}{49}\\ -\frac{29}{49}& -\frac{5}{49}& \frac{31}{49}& \frac{18}{49}\\ \frac{12}{49}& -\frac{3}{49}& -\frac{1}{49}& \frac{1}{49}\end{array}\right]$

Now, applying the formula $AX=B\iff X=A^{-1}B$ ,

  $\left[\begin{array}{cccc}1& -2& -1& 2\\ 3& -5& -2& -3\\ 2& -5& -2& -5\\ -1& 4& 4& 11\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ -1\\ 2\end{array}\right]\iff \left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{cccc}-\frac{24}{49}& \frac{55}{49}& -\frac{47}{49}& -\frac{2}{49}\\ -\frac{10}{49}& \frac{27}{49}& -\frac{40}{49}& -\frac{9}{49}\\ -\frac{29}{49}& -\frac{5}{49}& \frac{31}{49}& \frac{18}{49}\\ \frac{12}{49}& -\frac{3}{49}& -\frac{1}{49}& \frac{1}{49}\end{array}\right]\left[\begin{array}{c}0\\ 1\\ -1\\ 2\end{array}\right]$

Finding the solutions,

  

  $\begin{array}{l}\Rightarrow \left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}0+\frac{55}{49}+\frac{47}{49}-\frac{4}{49}\\ 0+\frac{27}{49}+\frac{40}{49}-\frac{18}{49}\\ 0-\frac{5}{49}-\frac{31}{49}+\frac{36}{49}\\ 0-\frac{3}{49}+\frac{1}{49}+\frac{2}{49}\end{array}\right]\\ \Rightarrow \left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}2\\ 1\\ 0\\ 0\end{array}\right]\end{array}$

Conclusion:

Therefore, the solution to given system of equations is $x_1=2,x_2=1,x_3=0,x_4=0.$