Linear Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify $R_{1}$ .

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

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 $0$ .

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

Step 2.2

Simplify $R_{2}$ .

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

Step 3

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

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Step 3.1

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

$[\begin{array}{c} 1 & - \frac{3}{2} & - 1 \\ 0 & 4 & 3 \\ 3 - 3 \cdot 1 & 2 - 3 (- \frac{3}{2}) & 1 - 3 \cdot - 1 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & - 1 \\ 0 & 4 & 3 \\ 0 & \frac{13}{2} & 4 \end{array}]$

Step 4

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

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Step 4.1

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

$[\begin{array}{c} 1 & - \frac{3}{2} & - 1 \\ \frac{0}{4} & \frac{4}{4} & \frac{3}{4} \\ 0 & \frac{13}{2} & 4 \end{array}]$

Step 4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & - 1 \\ 0 & 1 & \frac{3}{4} \\ 0 & \frac{13}{2} & 4 \end{array}]$

Step 5

Perform the row operation $R_{3} = R_{3} - \frac{13}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 5.1

Perform the row operation $R_{3} = R_{3} - \frac{13}{2} R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & - 1 \\ 0 & 1 & \frac{3}{4} \\ 0 - \frac{13}{2} \cdot 0 & \frac{13}{2} - \frac{13}{2} \cdot 1 & 4 - \frac{13}{2} \cdot \frac{3}{4} \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & - 1 \\ 0 & 1 & \frac{3}{4} \\ 0 & 0 & - \frac{7}{8} \end{array}]$

Step 6

Multiply each element of $R_{3}$ by $- \frac{8}{7}$ to make the entry at $3, 3$ a $1$ .

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Step 6.1

Multiply each element of $R_{3}$ by $- \frac{8}{7}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{3}{2} & - 1 \\ 0 & 1 & \frac{3}{4} \\ - \frac{8}{7} \cdot 0 & - \frac{8}{7} \cdot 0 & - \frac{8}{7} (- \frac{7}{8}) \end{array}]$

Step 6.2

Simplify $R_{3}$ .

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

Step 7

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

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Step 7.1

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

$[\begin{array}{c} 1 & - \frac{3}{2} & - 1 \\ 0 - \frac{3}{4} \cdot 0 & 1 - \frac{3}{4} \cdot 0 & \frac{3}{4} - \frac{3}{4} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Step 7.2

Simplify $R_{2}$ .

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify $R_{1}$ .

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

Step 9

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 9.1

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

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

Step 9.2

Simplify $R_{1}$ .

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