Linear Algebra Examples

$[\begin{array}{c} n & - 8 & 6 \\ 1 & 2 & 3 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} \frac{n}{n} & \frac{- 8}{n} & \frac{6}{n} \\ 1 & 2 & 3 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{8}{n} & \frac{6}{n} \\ 1 & 2 & 3 \end{array}]$

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 $0$ .

$[\begin{array}{c} 1 & - \frac{8}{n} & \frac{6}{n} \\ 1 - 1 & 2 + \frac{8}{n} & 3 - \frac{6}{n} \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{8}{n} & \frac{6}{n} \\ 0 & 2 + \frac{8}{n} & 3 - \frac{6}{n} \end{array}]$

Step 3

Multiply each element of $R_{2}$ by $\frac{1}{2 + \frac{8}{n}}$ to make the entry at $2, 2$ a $1$ .

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Step 3.1

Multiply each element of $R_{2}$ by $\frac{1}{2 + \frac{8}{n}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{8}{n} & \frac{6}{n} \\ \frac{0}{2 + \frac{8}{n}} & \frac{2 + \frac{8}{n}}{2 + \frac{8}{n}} & \frac{3 - \frac{6}{n}}{2 + \frac{8}{n}} \end{array}]$

Step 3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{8}{n} & \frac{6}{n} \\ 0 & 1 & \frac{3 (n - 2)}{2 (n + 4)} \end{array}]$

Step 4

Perform the row operation $R_{1} = R_{1} + \frac{8}{n} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 4.1

Perform the row operation $R_{1} = R_{1} + \frac{8}{n} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + \frac{8}{n} \cdot 0 & - \frac{8}{n} + \frac{8}{n} \cdot 1 & \frac{6}{n} + \frac{8}{n} \cdot \frac{3 (n - 2)}{2 (n + 4)} \\ 0 & 1 & \frac{3 (n - 2)}{2 (n + 4)} \end{array}]$

Step 4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{18}{n + 4} \\ 0 & 1 & \frac{3 (n - 2)}{2 (n + 4)} \end{array}]$