Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 4 & - 3 & 1 & 0 1 & 0 & - 2 & 0 - 2 & 1 & 1 & 0 \end{matrix} ⎤ ⎥ ⎦

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

Step 1

Multiply each element of

R_{1}

$R_{1}$ by

$\frac{1}{4}$ to make the entry at

$1, 1$ a

$1$ .

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Step 1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{4}$ to make the entry at

$1, 1$ a

$1$ .

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

Step 1.2

Simplify

$R_{1}$ .

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

Step 2

Perform the row operation

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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Step 2.1

Perform the row operation

$R_{2} = R_{2} - R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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

Step 2.2

Simplify

$R_{2}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ 0 & \frac{3}{4} & - \frac{9}{4} & 0 \\ - 2 & 1 & 1 & 0 \end{array}]$

Step 3

Perform the row operation

$R_{3} = R_{3} + 2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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Step 3.1

Perform the row operation

$R_{3} = R_{3} + 2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ 0 & \frac{3}{4} & - \frac{9}{4} & 0 \\ - 2 + 2 \cdot 1 & 1 + 2 (- \frac{3}{4}) & 1 + 2 (\frac{1}{4}) & 0 + 2 \cdot 0 \end{array}]$

Step 3.2

Simplify

$R_{3}$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ 0 & \frac{3}{4} & - \frac{9}{4} & 0 \\ 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{array}]$

Step 4

Multiply each element of

$R_{2}$ by

$\frac{4}{3}$ to make the entry at

$2, 2$ a

$1$ .

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Step 4.1

Multiply each element of

$R_{2}$ by

$\frac{4}{3}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & - \frac{3}{4} & \frac{1}{4} & 0 \\ \frac{4}{3} \cdot 0 & \frac{4}{3} \cdot \frac{3}{4} & \frac{4}{3} (- \frac{9}{4}) & \frac{4}{3} \cdot 0 \\ 0 & - \frac{1}{2} & \frac{3}{2} & 0 \end{array}]$

Step 4.2

Simplify

$R_{2}$ .

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

Step 5

Perform the row operation

$R_{3} = R_{3} + \frac{1}{2} R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Step 5.1

Perform the row operation

$R_{3} = R_{3} + \frac{1}{2} R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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

Step 5.2

Simplify

$R_{3}$ .

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

Step 6

Perform the row operation

$R_{1} = R_{1} + \frac{3}{4} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Step 6.1

Perform the row operation

$R_{1} = R_{1} + \frac{3}{4} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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

Step 6.2

Simplify

$R_{1}$ .

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