Linear Algebra Examples

$[\begin{array}{c} 0 & 1 & - 2 & 1 & 1 & 5 \\ - 2 & 3 & - 6 & 0 & 6 & 9 \\ 1 & 1 & - 2 & 0 & 2 & 3 \end{array}]$

Step 1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} - 2 & 3 & - 6 & 0 & 6 & 9 \\ 0 & 1 & - 2 & 1 & 1 & 5 \\ 1 & 1 & - 2 & 0 & 2 & 3 \end{array}]$

Step 2

Multiply each element of $R_{1}$ by $- \frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

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Step 2.1

Multiply each element of $R_{1}$ by $- \frac{1}{2}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} - \frac{1}{2} \cdot - 2 & - \frac{1}{2} \cdot 3 & - \frac{1}{2} \cdot - 6 & - \frac{1}{2} \cdot 0 & - \frac{1}{2} \cdot 6 & - \frac{1}{2} \cdot 9 \\ 0 & 1 & - 2 & 1 & 1 & 5 \\ 1 & 1 & - 2 & 0 & 2 & 3 \end{array}]$

Step 2.2

Simplify $R_{1}$ .

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

Step 3

Perform the row operation $R_{3} = R_{3} - R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 3.1

Perform the row operation $R_{3} = R_{3} - R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & 3 & 0 & - 3 & - \frac{9}{2} \\ 0 & 1 & - 2 & 1 & 1 & 5 \\ 1 - 1 & 1 + \frac{3}{2} & - 2 - 3 & 0 - 0 & 2 + 3 & 3 + \frac{9}{2} \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & 3 & 0 & - 3 & - \frac{9}{2} \\ 0 & 1 & - 2 & 1 & 1 & 5 \\ 0 & \frac{5}{2} & - 5 & 0 & 5 & \frac{15}{2} \end{array}]$

Step 4

Perform the row operation $R_{3} = R_{3} - \frac{5}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 4.1

Perform the row operation $R_{3} = R_{3} - \frac{5}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & 3 & 0 & - 3 & - \frac{9}{2} \\ 0 & 1 & - 2 & 1 & 1 & 5 \\ 0 - \frac{5}{2} \cdot 0 & \frac{5}{2} - \frac{5}{2} \cdot 1 & - 5 - \frac{5}{2} \cdot - 2 & 0 - \frac{5}{2} \cdot 1 & 5 - \frac{5}{2} \cdot 1 & \frac{15}{2} - \frac{5}{2} \cdot 5 \end{array}]$

Step 4.2

Simplify $R_{3}$ .

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

Step 5

Multiply each element of $R_{3}$ by $- \frac{2}{5}$ to make the entry at $3, 4$ a $1$ .

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Step 5.1

Multiply each element of $R_{3}$ by $- \frac{2}{5}$ to make the entry at $3, 4$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & 3 & 0 & - 3 & - \frac{9}{2} \\ 0 & 1 & - 2 & 1 & 1 & 5 \\ - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} \cdot 0 & - \frac{2}{5} (- \frac{5}{2}) & - \frac{2}{5} \cdot \frac{5}{2} & - \frac{2}{5} \cdot - 5 \end{array}]$

Step 5.2

Simplify $R_{3}$ .

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

Step 6

Perform the row operation $R_{2} = R_{2} - R_{3}$ to make the entry at $2, 4$ a $0$ .

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Step 6.1

Perform the row operation $R_{2} = R_{2} - R_{3}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & 3 & 0 & - 3 & - \frac{9}{2} \\ 0 - 0 & 1 - 0 & - 2 - 0 & 1 - 1 & 1 + 1 & 5 - 2 \\ 0 & 0 & 0 & 1 & - 1 & 2 \end{array}]$

Step 6.2

Simplify $R_{2}$ .

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

Step 7

Perform the row operation $R_{1} = R_{1} + \frac{3}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 7.1

Perform the row operation $R_{1} = R_{1} + \frac{3}{2} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{3}{2} \cdot 0 & - \frac{3}{2} + \frac{3}{2} \cdot 1 & 3 + \frac{3}{2} \cdot - 2 & 0 + \frac{3}{2} \cdot 0 & - 3 + \frac{3}{2} \cdot 2 & - \frac{9}{2} + \frac{3}{2} \cdot 3 \\ 0 & 1 & - 2 & 0 & 2 & 3 \\ 0 & 0 & 0 & 1 & - 1 & 2 \end{array}]$

Step 7.2

Simplify $R_{1}$ .

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