We may prefer to solve a linear system using matrix methods. To perform this action, we can set up an augmented matrix, and then use row operations to place our matrix in row-echelon form or reduced-row echelon form. At this point, we will be able to find our solution for the system.

Test Objectives
• Demonstrate the ability to set up an augmented matrix
• Demonstrate the ability to place a matrix in row-echelon form
• Demonstrate the ability to place a matrix in reduced-row echelon form
• Demonstrate the ability to solve a linear system using matrix methods
Gaussian Elimination Four Variables Practice Test:

#1:

Instructions: solve each system.

$$a)\hspace{.2em}$$ $$4x_{1}- 3x_{2}+ x_{3}- x_{4}=0$$ $$-2x_{1}+ 5x_{2}- 7x_{3}+ x_{4}=24$$ $$x_{1}- x_{2}+ 2x_{3}- 5x_{4}=3$$ $$3x_{1}+ 4x_{2}- 3x_{3}+ 2x_{4}=41$$

$$b)\hspace{.2em}$$ $$9x_{1}+ 2x_{2}- 3x_{3}+ x_{4}=15$$ $$-x_{1}+ 4x_{2}+ x_{3}- 2x_{4}=25$$ $$-3x_{1}- x_{2}+ 2x_{3}- 5x_{4}=-5$$ $$x_{1}+ x_{2}- 2x_{3}+ 5x_{4}=3$$

#2:

Instructions: solve each system.

$$a)\hspace{.2em}$$ $$x_{1}+ x_{2}+ x_{3}+ x_{4}=6$$ $$2x_{1}+ 3x_{2}+ 5x_{3}- x_{4}=15$$ $$-3x_{1}+ 4x_{2}+ x_{3}+ 2x_{4}=4$$ $$x_{1}+ 2x_{2}- x_{3}+ x_{4}=0$$

$$b)\hspace{.2em}$$ $$2x_{1}- 3x_{2}+ 5x_{3}+ x_{4}=11$$ $$-7x_{1}+ x_{2}- 2x_{3}- x_{4}=20$$ $$-x_{1}+ 4x_{2}- x_{3}+ 6x_{4}=3$$ $$x_{1}+ x_{2}+ x_{3}+ x_{4}=-1$$

#3:

Instructions: solve each system.

$$a)\hspace{.2em}$$ $$x_{1}+ x_{2}+ 5x_{3}+ 2x_{4}=26$$ $$7x_{1}- 2x_{2}+ x_{3}+ x_{4}=25$$ $$3x_{1}- 2x_{2}+ 4x_{3}- x_{4}=30$$ $$-x_{1}- x_{2}- x_{3}+ 9x_{4}=-17$$

#4:

Instructions: write in reduced-row echelon form.

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}4&1&-1&3&9\\ -1&1&-2&5&-7 \\ 3&7&-4&0&-9 \\ 2&-1&1&0&9 \end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}1&0&-3&2&20\\ 1&-2&1&4&28 \\ 0&-5&3&7&39 \\ 1&2&0&-8&-43 \end{array}\right]$$

#5:

Instructions: write in reduced-row echelon form.

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}-1&3&5&0&-26\\ 0&1&-10&2&55 \\ 1&0&20&4&-90 \\ 1&-3&0&0&1 \end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}-2&-3&0&4&6\\ 1&-7&1&0&-3 \\ 5&-1&-2&0&-1 \\ 0&3&-4&1&9 \end{array}\right]$$

$$c)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}1&0&-3&1&-2\\ -2&4&7&0&-6 \\ -3&0&0&5&-12 \\ 6&1&1&9&23 \end{array}\right]$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}(5,7,0,-1)$$

$$b)\hspace{.2em}(1,6,2,0)$$

#2:

Solutions:

$$a)\hspace{.2em}(1,0,3,2)$$

$$b)\hspace{.2em}(-4,-1,3,1)$$

#3:

Solutions:

$$a)\hspace{.2em}(3,0,5,-1)$$

#4:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}1&0&0&0&3\\ 0&1&0&0&-2 \\ 0&0&1&0&1 \\ 0&0&0&1&0 \end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}1&0&0&0&5\\ 0&1&0&0&0 \\ 0&0&1&0&-1 \\ 0&0&0&1&6 \end{array}\right]$$

#5:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}1&0&0&0&-2\\ 0&1&0&0&-1 \\ 0&0&1&0&-5 \\ 0&0&0&1&3 \end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}1&0&0&0&-1\\ 0&1&0&0&0 \\ 0&0&1&0&-2 \\ 0&0&0&1&1 \end{array}\right]$$

$$c)\hspace{.2em}$$ $$\left[ \begin{array}{cccc|c}1&0&0&0&4\\ 0&1&0&0&-3 \\ 0&0&1&0&2 \\ 0&0&0&1&0 \end{array}\right]$$