Linear Algebra Examples

$[\begin{array}{c} 3 & - 2 & 7 & 63 \\ 2 & 1 & - 21 & 28 \\ 1 & 4 & - 49 & - 7 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} \frac{3}{3} & \frac{- 2}{3} & \frac{7}{3} & \frac{63}{3} \\ 2 & 1 & - 21 & 28 \\ 1 & 4 & - 49 & - 7 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{7}{3} & 21 \\ 2 & 1 & - 21 & 28 \\ 1 & 4 & - 49 & - 7 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} - 2 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{7}{3} & 21 \\ 2 - 2 \cdot 1 & 1 - 2 (- \frac{2}{3}) & - 21 - 2 (\frac{7}{3}) & 28 - 2 \cdot 21 \\ 1 & 4 & - 49 & - 7 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{7}{3} & 21 \\ 0 & \frac{7}{3} & - \frac{77}{3} & - 14 \\ 1 & 4 & - 49 & - 7 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{7}{3} & 21 \\ 0 & \frac{7}{3} & - \frac{77}{3} & - 14 \\ 1 - 1 & 4 + \frac{2}{3} & - 49 - \frac{7}{3} & - 7 - 21 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{7}{3} & 21 \\ 0 & \frac{7}{3} & - \frac{77}{3} & - 14 \\ 0 & \frac{14}{3} & - \frac{154}{3} & - 28 \end{array}]$

Step 4

Multiply each element of $R_{2}$ by $\frac{3}{7}$ to make the entry at $2, 2$ a $1$ .

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Step 4.1

Multiply each element of $R_{2}$ by $\frac{3}{7}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{7}{3} & 21 \\ \frac{3}{7} \cdot 0 & \frac{3}{7} \cdot \frac{7}{3} & \frac{3}{7} (- \frac{77}{3}) & \frac{3}{7} \cdot - 14 \\ 0 & \frac{14}{3} & - \frac{154}{3} & - 28 \end{array}]$

Step 4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{7}{3} & 21 \\ 0 & 1 & - 11 & - 6 \\ 0 & \frac{14}{3} & - \frac{154}{3} & - 28 \end{array}]$

Step 5

Perform the row operation $R_{3} = R_{3} - \frac{14}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 5.1

Perform the row operation $R_{3} = R_{3} - \frac{14}{3} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{7}{3} & 21 \\ 0 & 1 & - 11 & - 6 \\ 0 - \frac{14}{3} \cdot 0 & \frac{14}{3} - \frac{14}{3} \cdot 1 & - \frac{154}{3} - \frac{14}{3} \cdot - 11 & - 28 - \frac{14}{3} \cdot - 6 \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2}{3} & \frac{7}{3} & 21 \\ 0 & 1 & - 11 & - 6 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 6

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

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Step 6.1

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

$[\begin{array}{c} 1 + \frac{2}{3} \cdot 0 & - \frac{2}{3} + \frac{2}{3} \cdot 1 & \frac{7}{3} + \frac{2}{3} \cdot - 11 & 21 + \frac{2}{3} \cdot - 6 \\ 0 & 1 & - 11 & - 6 \\ 0 & 0 & 0 & 0 \end{array}]$

Step 6.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 5 & 17 \\ 0 & 1 & - 11 & - 6 \\ 0 & 0 & 0 & 0 \end{array}]$