Linear Algebra Examples

$[\begin{array}{c} - 3 & 6 & 21 & 30 \\ 1 & 4 & 9 & 12 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} - \frac{1}{3} \cdot - 3 & - \frac{1}{3} \cdot 6 & - \frac{1}{3} \cdot 21 & - \frac{1}{3} \cdot 30 \\ 1 & 4 & 9 & 12 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & - 7 & - 10 \\ 1 & 4 & 9 & 12 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - 2 & - 7 & - 10 \\ 1 - 1 & 4 + 2 & 9 + 7 & 12 + 10 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 7 & - 10 \\ 0 & 6 & 16 & 22 \end{array}]$

Step 3

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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Step 3.1

Multiply each element of $R_{2}$ by $\frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 2 & - 7 & - 10 \\ \frac{0}{6} & \frac{6}{6} & \frac{16}{6} & \frac{22}{6} \end{array}]$

Step 3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 7 & - 10 \\ 0 & 1 & \frac{8}{3} & \frac{11}{3} \end{array}]$

Step 4

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

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Step 4.1

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

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 7 + 2 (\frac{8}{3}) & - 10 + 2 (\frac{11}{3}) \\ 0 & 1 & \frac{8}{3} & \frac{11}{3} \end{array}]$

Step 4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - \frac{5}{3} & - \frac{8}{3} \\ 0 & 1 & \frac{8}{3} & \frac{11}{3} \end{array}]$