Linear Algebra Examples

$[\begin{array}{c} 1 & - 4 & - 6 & - 1 & 7 & 0 \\ 0 & 0 & 1 & 1 & - 8 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} 1 & - 4 & - 6 & - 1 & 7 & 0 \\ 0 & 0 & 1 & 1 & - 8 & 0 \\ \frac{0}{2} & \frac{0}{2} & \frac{0}{2} & \frac{0}{2} & \frac{2}{2} & \frac{0}{2} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 1.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 4 & - 6 & - 1 & 7 & 0 \\ 0 & 0 & 1 & 1 & - 8 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} + 8 R_{3}$ to make the entry at $2, 5$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} + 8 R_{3}$ to make the entry at $2, 5$ a $0$ .

$[\begin{array}{c} 1 & - 4 & - 6 & - 1 & 7 & 0 \\ 0 + 8 \cdot 0 & 0 + 8 \cdot 0 & 1 + 8 \cdot 0 & 1 + 8 \cdot 0 & - 8 + 8 \cdot 1 & 0 + 8 \cdot 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 4 & - 6 & - 1 & 7 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 3

Perform the row operation $R_{1} = R_{1} - 7 R_{3}$ to make the entry at $1, 5$ a $0$ .

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Step 3.1

Perform the row operation $R_{1} = R_{1} - 7 R_{3}$ to make the entry at $1, 5$ a $0$ .

$[\begin{array}{c} 1 - 7 \cdot 0 & - 4 - 7 \cdot 0 & - 6 - 7 \cdot 0 & - 1 - 7 \cdot 0 & 7 - 7 \cdot 1 & 0 - 7 \cdot 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 4 & - 6 & - 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 4

Perform the row operation $R_{1} = R_{1} + 6 R_{2}$ to make the entry at $1, 3$ a $0$ .

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Step 4.1

Perform the row operation $R_{1} = R_{1} + 6 R_{2}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 6 \cdot 0 & - 4 + 6 \cdot 0 & - 6 + 6 \cdot 1 & - 1 + 6 \cdot 1 & 0 + 6 \cdot 0 & 0 + 6 \cdot 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 4 & 0 & 5 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$