Linear Algebra Examples

$[\begin{array}{c} 2 & 0 & 6 \\ - 3 & 5 & - 19 \\ 4 & 1 & 10 \\ 1 & - 2 & 7 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} \frac{2}{2} & \frac{0}{2} & \frac{6}{2} \\ - 3 & 5 & - 19 \\ 4 & 1 & 10 \\ 1 & - 2 & 7 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ - 3 & 5 & - 19 \\ 4 & 1 & 10 \\ 1 & - 2 & 7 \end{array}]$

Step 2

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

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Step 2.1

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

$[\begin{array}{c} 1 & 0 & 3 \\ - 3 + 3 \cdot 1 & 5 + 3 \cdot 0 & - 19 + 3 \cdot 3 \\ 4 & 1 & 10 \\ 1 & - 2 & 7 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 5 & - 10 \\ 4 & 1 & 10 \\ 1 & - 2 & 7 \end{array}]$

Step 3

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

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Step 3.1

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

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 5 & - 10 \\ 4 - 4 \cdot 1 & 1 - 4 \cdot 0 & 10 - 4 \cdot 3 \\ 1 & - 2 & 7 \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 5 & - 10 \\ 0 & 1 & - 2 \\ 1 & - 2 & 7 \end{array}]$

Step 4

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

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Step 4.1

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

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 5 & - 10 \\ 0 & 1 & - 2 \\ 1 - 1 & - 2 - 0 & 7 - 3 \end{array}]$

Step 4.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 3 \\ 0 & 5 & - 10 \\ 0 & 1 & - 2 \\ 0 & - 2 & 4 \end{array}]$

Step 5

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

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Step 5.1

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

$[\begin{array}{c} 1 & 0 & 3 \\ \frac{0}{5} & \frac{5}{5} & \frac{- 10}{5} \\ 0 & 1 & - 2 \\ 0 & - 2 & 4 \end{array}]$

Step 5.2

Simplify $R_{2}$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify $R_{3}$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify $R_{4}$ .

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