Linear Algebra Examples

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

Step 1

Swap $R_{4}$ with $R_{1}$ to put a nonzero entry at $1, 2$ .

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

Step 2

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 3$ .

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

Step 3

Perform the row operation $R_{2} = R_{2} - R_{4}$ to make the entry at $2, 5$ a $0$ .

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Step 3.1

Perform the row operation $R_{2} = R_{2} - R_{4}$ to make the entry at $2, 5$ a $0$ .

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

Step 3.2

Simplify $R_{2}$ .

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

Step 4

Perform the row operation $R_{1} = R_{1} - R_{4}$ to make the entry at $1, 5$ a $0$ .

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Step 4.1

Perform the row operation $R_{1} = R_{1} - R_{4}$ to make the entry at $1, 5$ a $0$ .

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

Step 4.2

Simplify $R_{1}$ .

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

Step 5

Perform the row operation $R_{1} = R_{1} - R_{3}$ to make the entry at $1, 4$ a $0$ .

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Step 5.1

Perform the row operation $R_{1} = R_{1} - R_{3}$ to make the entry at $1, 4$ a $0$ .

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

Step 5.2

Simplify $R_{1}$ .

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