Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} a & a & 1 - a & 1 a & a^{2} a & 1 - a^{2} & 1 2 a & a + a^{2} & 2 - 2 a & a + 2 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} a & a & 1 - a & 1 \\ a & a^{2} a & 1 - a^{2} & 1 \\ 2 a & a + a^{2} & 2 - 2 a & a + 2 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

Multiply

$a^{2}$ by

$a$ by adding the exponents.

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Step 1.1.1

Multiply

$a^{2}$ by

$a$ .

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Step 1.1.1.1

Raise

$a$ to the power of

$1$ .

$[\begin{array}{c} a & a & 1 - a & 1 \\ a & a^{2} a^{1} & 1 - a^{2} & 1 \\ 2 a & a + a^{2} & 2 - 2 a & a + 2 \end{array}]$

Step 1.1.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$[\begin{array}{c} a & a & 1 - a & 1 \\ a & a^{2 + 1} & 1 - a^{2} & 1 \\ 2 a & a + a^{2} & 2 - 2 a & a + 2 \end{array}]$

Step 1.1.2

Add

$2$ and

$1$ .

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

Step 1.2

Multiply each element of

$R_{1}$ by

$\frac{1}{a}$ to make the entry at

$1, 1$ a

$1$ .

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Step 1.2.1

Multiply each element of

$R_{1}$ by

$\frac{1}{a}$ to make the entry at

$1, 1$ a

$1$ .

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

Step 1.2.2

Simplify

$R_{1}$ .

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

Step 1.3

Perform the row operation

$R_{2} = R_{2} - a R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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Step 1.3.1

Perform the row operation

$R_{2} = R_{2} - a R_{1}$ to make the entry at

$2, 1$ a

$0$ .

$[\begin{array}{c} 1 & 1 & \frac{1 - a}{a} & \frac{1}{a} \\ a - a \cdot 1 & a^{3} - a \cdot 1 & 1 - a^{2} - a \frac{1 - a}{a} & 1 - a \frac{1}{a} \\ 2 a & a + a^{2} & 2 - 2 a & a + 2 \end{array}]$

Step 1.3.2

Simplify

$R_{2}$ .

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

Step 1.4

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 1.4.1

Perform the row operation

$R_{3} = R_{3} - 2 a R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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

Step 1.4.2

Simplify

$R_{3}$ .

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

Step 1.5

Multiply each element of

$R_{2}$ by

$\frac{1}{a^{3} - a}$ to make the entry at

$2, 2$ a

$1$ .

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Step 1.5.1

Multiply each element of

$R_{2}$ by

$\frac{1}{a^{3} - a}$ to make the entry at

$2, 2$ a

$1$ .

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

Step 1.5.2

Simplify

$R_{2}$ .

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

Step 1.6

Perform the row operation

$R_{3} = R_{3} - (a^{2} - a) R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Step 1.6.1

Perform the row operation

$R_{3} = R_{3} - (a^{2} - a) R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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

Step 1.6.2

Simplify

$R_{3}$ .

$[\begin{array}{c} 1 & 1 & \frac{1 - a}{a} & \frac{1}{a} \\ 0 & 1 & - \frac{1}{a + 1} & 0 \\ 0 & 0 & \frac{a (a - 1)}{a + 1} & a \end{array}]$

Step 1.7

Multiply each element of

$R_{3}$ by

$\frac{a + 1}{a (a - 1)}$ to make the entry at

$3, 3$ a

$1$ .

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Step 1.7.1

Multiply each element of

$R_{3}$ by

$\frac{a + 1}{a (a - 1)}$ to make the entry at

$3, 3$ a

$1$ .

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

Step 1.7.2

Simplify

$R_{3}$ .

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

Step 1.8

Perform the row operation

$R_{2} = R_{2} + \frac{1}{a + 1} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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Step 1.8.1

Perform the row operation

$R_{2} = R_{2} + \frac{1}{a + 1} R_{3}$ to make the entry at

$2, 3$ a

$0$ .

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

Step 1.8.2

Simplify

$R_{2}$ .

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

Step 1.9

Perform the row operation

$R_{1} = R_{1} - \frac{1 - a}{a} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

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Step 1.9.1

Perform the row operation

$R_{1} = R_{1} - \frac{1 - a}{a} R_{3}$ to make the entry at

$1, 3$ a

$0$ .

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

Step 1.9.2

Simplify

$R_{1}$ .

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

Step 1.10

Perform the row operation

$R_{1} = R_{1} - R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Step 1.10.1

Perform the row operation

$R_{1} = R_{1} - R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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

Step 1.10.2

Simplify

$R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{a^{2} - 2}{a (a - 1)} \\ 0 & 1 & 0 & \frac{1}{a - 1} \\ 0 & 0 & 1 & \frac{a + 1}{a - 1} \end{array}]$

Step 2

The pivot positions are the locations with the leading

$1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions:

$a_{11}, a_{22},$ and

$a_{33}$

Pivot Columns:

$1, 2,$ and

$3$