Linear Algebra Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

Simplify $R_{1}$ .

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify $R_{2}$ .

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify $R_{3}$ .

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

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

Simplify $R_{2}$ .

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

Step 5

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

Tap for more steps...

Step 5.1

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

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

Step 5.2

Simplify $R_{3}$ .

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

Step 6

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

Tap for more steps...

Step 6.1

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

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

Step 6.2

Simplify $R_{2}$ .

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

Step 7

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

Tap for more steps...

Step 7.1

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

$[\begin{array}{c} 1 - (- 1 + \sqrt{2}) \cdot 0 & 0 - (- 1 + \sqrt{2}) \cdot 0 & - 1 + \sqrt{2} - (- 1 + \sqrt{2}) \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 7.2

Simplify $R_{1}$ .

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