Linear Algebra Examples

$[\begin{array}{c} a & - 2 b & 3 d = 11 \\ 4 a & b & - d = 4 \\ 2 a & - b & 3 d = 10 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} \frac{a}{a} & \frac{- 2 b}{a} & \frac{3 d = 11}{a} \\ 4 a & b & - d = 4 \\ 2 a & - b & 3 d = 10 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 4 a & b & - d = 4 \\ 2 a & - b & 3 d = 10 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} - 4 a R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} - 4 a R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 4 a - 4 a \cdot 1 & b - 4 a (- \frac{2 b}{a}) & - d = 4 - 4 a \frac{3 d = 11}{a} \\ 2 a & - b & 3 d = 10 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 & 9 b & - d = 4 - 43 d = 11 \\ 2 a & - b & 3 d = 10 \end{array}]$

Step 3

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

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Step 3.1

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

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 & 9 b & - d = 4 - 43 d = 11 \\ 2 a - 2 a \cdot 1 & - b - 2 a (- \frac{2 b}{a}) & 3 d = 10 - 2 a \frac{3 d = 11}{a} \end{array}]$

Step 3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 & 9 b & - d = 4 - 43 d = 11 \\ 0 & 3 b & 3 d = 10 - 23 d = 11 \end{array}]$

Step 4

Multiply each element of $R_{2}$ by $\frac{1}{9 b}$ to make the entry at $2, 2$ a $1$ .

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Step 4.1

Multiply each element of $R_{2}$ by $\frac{1}{9 b}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ \frac{0}{9 b} & \frac{9 b}{9 b} & \frac{- d = 4 - 43 d = 11}{9 b} \\ 0 & 3 b & 3 d = 10 - 23 d = 11 \end{array}]$

Step 4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 & 1 & \frac{- d = 4 - 43 d = 11}{9 b} \\ 0 & 3 b & 3 d = 10 - 23 d = 11 \end{array}]$

Step 5

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

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Step 5.1

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

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 & 1 & \frac{- d = 4 - 43 d = 11}{9 b} \\ 0 - 3 b \cdot 0 & 3 b - 3 b \cdot 1 & 3 d = 10 - 23 d = 11 - 3 b \frac{- d = 4 - 43 d = 11}{9 b} \end{array}]$

Step 5.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 & 1 & \frac{- d = 4 - 43 d = 11}{9 b} \\ 0 & 0 & \frac{33 d = 10 - - d = 4 - 23 d = 11}{3} \end{array}]$

Step 6

Multiply each element of $R_{3}$ by $\frac{3}{33 d = 10 - - d = 4 - 23 d = 11}$ to make the entry at $3, 3$ a $1$ .

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Step 6.1

Multiply each element of $R_{3}$ by $\frac{3}{33 d = 10 - - d = 4 - 23 d = 11}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 & 1 & \frac{- d = 4 - 43 d = 11}{9 b} \\ \frac{3}{33 d = 10 - - d = 4 - 23 d = 11} \cdot 0 & \frac{3}{33 d = 10 - - d = 4 - 23 d = 11} \cdot 0 & \frac{3}{33 d = 10 - - d = 4 - 23 d = 11} \cdot \frac{33 d = 10 - - d = 4 - 23 d = 11}{3} \end{array}]$

Step 6.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 & 1 & \frac{- d = 4 - 43 d = 11}{9 b} \\ 0 & 0 & 1 \end{array}]$

Step 7

Perform the row operation $R_{2} = R_{2} - \frac{- d = 4 - 43 d = 11}{9 b} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 7.1

Perform the row operation $R_{2} = R_{2} - \frac{- d = 4 - 43 d = 11}{9 b} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 - \frac{- d = 4 - 43 d = 11}{9 b} \cdot 0 & 1 - \frac{- d = 4 - 43 d = 11}{9 b} \cdot 0 & \frac{- d = 4 - 43 d = 11}{9 b} - \frac{- d = 4 - 43 d = 11}{9 b} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Step 7.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & \frac{3 d = 11}{a} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 8

Perform the row operation $R_{1} = R_{1} - \frac{3 d = 11}{a} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 8.1

Perform the row operation $R_{1} = R_{1} - \frac{3 d = 11}{a} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{3 d = 11}{a} \cdot 0 & - \frac{2 b}{a} - \frac{3 d = 11}{a} \cdot 0 & \frac{3 d = 11}{a} - \frac{3 d = 11}{a} \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 8.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - \frac{2 b}{a} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 9

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

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Step 9.1

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

$[\begin{array}{c} 1 + \frac{2 b}{a} \cdot 0 & - \frac{2 b}{a} + \frac{2 b}{a} \cdot 1 & 0 + \frac{2 b}{a} \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 9.2

Simplify $R_{1}$ .

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