Linear Algebra Examples

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 1.1.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 5 - 5 \cdot 1 & 6 - 5 \cdot 2 & 7 - 5 \cdot 3 & 8 - 5 \cdot 4 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}]$

Step 1.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 4 & - 8 & - 12 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{array}]$

Step 1.2

Perform the row operation $R_{3} = R_{3} - 9 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 1.2.1

Perform the row operation $R_{3} = R_{3} - 9 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 4 & - 8 & - 12 \\ 9 - 9 \cdot 1 & 10 - 9 \cdot 2 & 11 - 9 \cdot 3 & 12 - 9 \cdot 4 \\ 13 & 14 & 15 & 16 \end{array}]$

Step 1.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 4 & - 8 & - 12 \\ 0 & - 8 & - 16 & - 24 \\ 13 & 14 & 15 & 16 \end{array}]$

Step 1.3

Perform the row operation $R_{4} = R_{4} - 13 R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 1.3.1

Perform the row operation $R_{4} = R_{4} - 13 R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 4 & - 8 & - 12 \\ 0 & - 8 & - 16 & - 24 \\ 13 - 13 \cdot 1 & 14 - 13 \cdot 2 & 15 - 13 \cdot 3 & 16 - 13 \cdot 4 \end{array}]$

Step 1.3.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & - 4 & - 8 & - 12 \\ 0 & - 8 & - 16 & - 24 \\ 0 & - 12 & - 24 & - 36 \end{array}]$

Step 1.4

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

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Step 1.4.1

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

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ - \frac{1}{4} \cdot 0 & - \frac{1}{4} \cdot - 4 & - \frac{1}{4} \cdot - 8 & - \frac{1}{4} \cdot - 12 \\ 0 & - 8 & - 16 & - 24 \\ 0 & - 12 & - 24 & - 36 \end{array}]$

Step 1.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & - 8 & - 16 & - 24 \\ 0 & - 12 & - 24 & - 36 \end{array}]$

Step 1.5

Perform the row operation $R_{3} = R_{3} + 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 1.5.1

Perform the row operation $R_{3} = R_{3} + 8 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 + 8 \cdot 0 & - 8 + 8 \cdot 1 & - 16 + 8 \cdot 2 & - 24 + 8 \cdot 3 \\ 0 & - 12 & - 24 & - 36 \end{array}]$

Step 1.5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & - 12 & - 24 & - 36 \end{array}]$

Step 1.6

Perform the row operation $R_{4} = R_{4} + 12 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 1.6.1

Perform the row operation $R_{4} = R_{4} + 12 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 + 12 \cdot 0 & - 12 + 12 \cdot 1 & - 24 + 12 \cdot 2 & - 36 + 12 \cdot 3 \end{array}]$

Step 1.6.2

Simplify $R_{4}$ .

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

Step 1.7

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

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Step 1.7.1

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

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

Step 1.7.2

Simplify $R_{1}$ .

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

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: $a_{11}$ and $a_{22}$

Pivot Columns: $1$ and $2$