Linear Algebra Examples

$[\begin{array}{c} 0 & 0 & 3 & 0 \\ 100000 & 40000 & 18000 & 254000 \end{array}]$

Step 1

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

$[\begin{array}{c} 100000 & 40000 & 18000 & 254000 \\ 0 & 0 & 3 & 0 \end{array}]$

Step 2

Multiply each element of $R_{1}$ by $\frac{1}{100000}$ to make the entry at $1, 1$ a $1$ .

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Step 2.1

Multiply each element of $R_{1}$ by $\frac{1}{100000}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{100000}{100000} & \frac{40000}{100000} & \frac{18000}{100000} & \frac{254000}{100000} \\ 0 & 0 & 3 & 0 \end{array}]$

Step 2.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{9}{50} & \frac{127}{50} \\ 0 & 0 & 3 & 0 \end{array}]$

Step 3

Multiply each element of $R_{2}$ by $\frac{1}{3}$ to make the entry at $2, 3$ a $1$ .

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Step 3.1

Multiply each element of $R_{2}$ by $\frac{1}{3}$ to make the entry at $2, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{9}{50} & \frac{127}{50} \\ \frac{0}{3} & \frac{0}{3} & \frac{3}{3} & \frac{0}{3} \end{array}]$

Step 3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & \frac{9}{50} & \frac{127}{50} \\ 0 & 0 & 1 & 0 \end{array}]$

Step 4

Perform the row operation $R_{1} = R_{1} - \frac{9}{50} R_{2}$ to make the entry at $1, 3$ a $0$ .

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Step 4.1

Perform the row operation $R_{1} = R_{1} - \frac{9}{50} R_{2}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{9}{50} \cdot 0 & \frac{2}{5} - \frac{9}{50} \cdot 0 & \frac{9}{50} - \frac{9}{50} \cdot 1 & \frac{127}{50} - \frac{9}{50} \cdot 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{2}{5} & 0 & \frac{127}{50} \\ 0 & 0 & 1 & 0 \end{array}]$