Linear Algebra Examples

$(u x v) \cdot w = [\begin{array}{c} a & b & d \\ g & h & j \\ k & l & n \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

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

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Step 1.1.1

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

$[\begin{array}{c} \frac{a}{a} & \frac{b}{a} & \frac{d}{a} \\ g & h & j \\ k & l & n \end{array}]$

Step 1.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ g & h & j \\ k & l & n \end{array}]$

Step 1.2

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

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Step 1.2.1

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

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ g - g \cdot 1 & h - g \frac{b}{a} & j - g \frac{d}{a} \\ k & l & n \end{array}]$

Step 1.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & h - \frac{b g}{a} & j - \frac{d g}{a} \\ k & l & n \end{array}]$

Step 1.3

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

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Step 1.3.1

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

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & h - \frac{b g}{a} & j - \frac{d g}{a} \\ k - k \cdot 1 & l - k \frac{b}{a} & n - k \frac{d}{a} \end{array}]$

Step 1.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & h - \frac{b g}{a} & j - \frac{d g}{a} \\ 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{array}]$

Step 1.4

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

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Step 1.4.1

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

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ \frac{0}{h - \frac{b g}{a}} & \frac{h - \frac{b g}{a}}{h - \frac{b g}{a}} & \frac{j - \frac{d g}{a}}{h - \frac{b g}{a}} \\ 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{array}]$

Step 1.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{array}]$

Step 1.5

Perform the row operation $R_{3} = R_{3} - (l - \frac{b k}{a}) R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 1.5.1

Perform the row operation $R_{3} = R_{3} - (l - \frac{b k}{a}) R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 - (l - \frac{b k}{a}) \cdot 0 & l - \frac{b k}{a} - (l - \frac{b k}{a}) \cdot 1 & n - \frac{d k}{a} - (l - \frac{b k}{a}) \frac{a j - d g}{a h - b g} \end{array}]$

Step 1.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 & 0 & - \frac{d k h - n a h + n b g + l a j - l d g - b k j}{a h - b g} \end{array}]$

Step 1.6

Multiply each element of $R_{3}$ by $- \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j}$ to make the entry at $3, 3$ a $1$ .

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Step 1.6.1

Multiply each element of $R_{3}$ by $- \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ - \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j} \cdot 0 & - \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j} \cdot 0 & - \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j} (- \frac{d k h - n a h + n b g + l a j - l d g - b k j}{a h - b g}) \end{array}]$

Step 1.6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 & 0 & 1 \end{array}]$

Step 1.7

Perform the row operation $R_{2} = R_{2} - \frac{a j - d g}{a h - b g} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 1.7.1

Perform the row operation $R_{2} = R_{2} - \frac{a j - d g}{a h - b g} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 - \frac{a j - d g}{a h - b g} \cdot 0 & 1 - \frac{a j - d g}{a h - b g} \cdot 0 & \frac{a j - d g}{a h - b g} - \frac{a j - d g}{a h - b g} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Step 1.7.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 1.8

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

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Step 1.8.1

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

$[\begin{array}{c} 1 - \frac{d}{a} \cdot 0 & \frac{b}{a} - \frac{d}{a} \cdot 0 & \frac{d}{a} - \frac{d}{a} \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 1.8.2

Simplify $R_{1}$ .

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

Step 1.9

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

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Step 1.9.1

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

$[\begin{array}{c} 1 - \frac{b}{a} \cdot 0 & \frac{b}{a} - \frac{b}{a} \cdot 1 & 0 - \frac{b}{a} \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 1.9.2

Simplify $R_{1}$ .

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

Step 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}, a_{22},$ and $a_{33}$

Pivot Columns: $1, 2,$ and $3$

Step 3

The rank is the number of pivot columns.

$3$