Linear Algebra Examples

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0.04 & 0.05 & 0.08 & 0.02 \\ 0.04 & - 0.1 & 0 & 0 \end{array}]$

Step 1

Perform the row operation $R_{2} = R_{2} - 0.04 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 1.1

Perform the row operation $R_{2} = R_{2} - 0.04 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0.04 - 0.04 \cdot 1 & 0.05 - 0.04 \cdot 1 & 0.08 - 0.04 \cdot 1 & 0.02 - 0.04 \cdot 1 \\ 0.04 & - 0.1 & 0 & 0 \end{array}]$

Step 1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 0.01 & 0.04 & - 0.02 \\ 0.04 & - 0.1 & 0 & 0 \end{array}]$

Step 2

Perform the row operation $R_{3} = R_{3} - 0.04 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{3} = R_{3} - 0.04 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 0.01 & 0.04 & - 0.02 \\ 0.04 - 0.04 \cdot 1 & - 0.1 - 0.04 \cdot 1 & 0 - 0.04 \cdot 1 & 0 - 0.04 \cdot 1 \end{array}]$

Step 2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 0.01 & 0.04 & - 0.02 \\ 0 & - 0.14 & - 0.04 & - 0.04 \end{array}]$

Step 3

Multiply each element of $R_{2}$ by $\frac{1}{0.01}$ to make the entry at $2, 2$ a $1$ .

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Step 3.1

Multiply each element of $R_{2}$ by $\frac{1}{0.01}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ \frac{0}{0.01} & \frac{0.01}{0.01} & \frac{0.04}{0.01} & \frac{- 0.02}{0.01} \\ 0 & - 0.14 & - 0.04 & - 0.04 \end{array}]$

Step 3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 4 & - 2 \\ 0 & - 0.14 & - 0.04 & - 0.04 \end{array}]$

Step 4

Perform the row operation $R_{3} = R_{3} + 0.14 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 4.1

Perform the row operation $R_{3} = R_{3} + 0.14 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 4 & - 2 \\ 0 + 0.14 \cdot 0 & - 0.14 + 0.14 \cdot 1 & - 0.04 + 0.14 \cdot 4 & - 0.04 + 0.14 \cdot - 2 \end{array}]$

Step 4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 4 & - 2 \\ 0 & 0 & 0.52 & - 0.32 \end{array}]$

Step 5

Multiply each element of $R_{3}$ by $\frac{1}{0.52}$ to make the entry at $3, 3$ a $1$ .

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Step 5.1

Multiply each element of $R_{3}$ by $\frac{1}{0.52}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 4 & - 2 \\ \frac{0}{0.52} & \frac{0}{0.52} & \frac{0.52}{0.52} & \frac{- 0.32}{0.52} \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 4 & - 2 \\ 0 & 0 & 1 & - 0.\overline{615384} \end{array}]$

Step 6

Perform the row operation $R_{2} = R_{2} - 4 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 6.1

Perform the row operation $R_{2} = R_{2} - 4 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 - 4 \cdot 0 & 1 - 4 \cdot 0 & 4 - 4 \cdot 1 & - 2 - 4 \cdot - 0.\overline{615384} \\ 0 & 0 & 1 & - 0.\overline{615384} \end{array}]$

Step 6.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0.\overline{461538} \\ 0 & 0 & 1 & - 0.\overline{615384} \end{array}]$

Step 7

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

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Step 7.1

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

$[\begin{array}{c} 1 - 0 & 1 - 0 & 1 - 1 & 1 + 0.\overline{615384} \\ 0 & 1 & 0 & 0.\overline{461538} \\ 0 & 0 & 1 & - 0.\overline{615384} \end{array}]$

Step 7.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 0 & 1.\overline{615384} \\ 0 & 1 & 0 & 0.\overline{461538} \\ 0 & 0 & 1 & - 0.\overline{615384} \end{array}]$

Step 8

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

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Step 8.1

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

$[\begin{array}{c} 1 - 0 & 1 - 1 & 0 - 0 & 1.\overline{615384} - 0.\overline{461538} \\ 0 & 1 & 0 & 0.\overline{461538} \\ 0 & 0 & 1 & - 0.\overline{615384} \end{array}]$

Step 8.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1.\overline{153846} \\ 0 & 1 & 0 & 0.\overline{461538} \\ 0 & 0 & 1 & - 0.\overline{615384} \end{array}]$