Linear Algebra Examples

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

Step 1

Find the determinant.

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Step 1.1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 1.1.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 1.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 1.1.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} f & 1 \\ 1 & f \end{array} |$

Step 1.1.4

Multiply element $a_{11}$ by its cofactor.

$f | \begin{array}{c} f & 1 \\ 1 & f \end{array} |$

Step 1.1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 1 \\ 0 & f \end{array} |$

Step 1.1.6

Multiply element $a_{12}$ by its cofactor.

$- 1 | \begin{array}{c} 1 & 1 \\ 0 & f \end{array} |$

Step 1.1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & f \\ 0 & 1 \end{array} |$

Step 1.1.8

Multiply element $a_{13}$ by its cofactor.

$0 | \begin{array}{c} 1 & f \\ 0 & 1 \end{array} |$

Step 1.1.9

Add the terms together.

$f | \begin{array}{c} f & 1 \\ 1 & f \end{array} | - 1 | \begin{array}{c} 1 & 1 \\ 0 & f \end{array} | + 0 | \begin{array}{c} 1 & f \\ 0 & 1 \end{array} |$

Step 1.2

Multiply $0$ by $| \begin{array}{c} 1 & f \\ 0 & 1 \end{array} |$ .

$f | \begin{array}{c} f & 1 \\ 1 & f \end{array} | - 1 | \begin{array}{c} 1 & 1 \\ 0 & f \end{array} | + 0$

Step 1.3

Evaluate $| \begin{array}{c} f & 1 \\ 1 & f \end{array} |$ .

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Step 1.3.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$f (f \cdot f - 1 \cdot 1) - 1 | \begin{array}{c} 1 & 1 \\ 0 & f \end{array} | + 0$

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

Multiply $f$ by $f$ .

$f (f^{2} - 1 \cdot 1) - 1 | \begin{array}{c} 1 & 1 \\ 0 & f \end{array} | + 0$

Step 1.3.2.2

Multiply $- 1$ by $1$ .

$f (f^{2} - 1) - 1 | \begin{array}{c} 1 & 1 \\ 0 & f \end{array} | + 0$

Step 1.4

Evaluate $| \begin{array}{c} 1 & 1 \\ 0 & f \end{array} |$ .

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Step 1.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$f (f^{2} - 1) - 1 (1 f + 0 \cdot 1) + 0$

Step 1.4.2

Simplify the determinant.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Multiply $f$ by $1$ .

$f (f^{2} - 1) - 1 (f + 0 \cdot 1) + 0$

Step 1.4.2.1.2

Multiply $0$ by $1$ .

$f (f^{2} - 1) - 1 (f + 0) + 0$

Step 1.4.2.2

Add $f$ and $0$ .

$f (f^{2} - 1) - 1 f + 0$

Step 1.5

Simplify the determinant.

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Step 1.5.1

Add $f (f^{2} - 1) - 1 f$ and $0$ .

$f (f^{2} - 1) - 1 f$

Step 1.5.2

Simplify each term.

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Step 1.5.2.1

Apply the distributive property.

$f \cdot f^{2} + f \cdot - 1 - 1 f$

Step 1.5.2.2

Multiply $f$ by $f^{2}$ by adding the exponents.

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Step 1.5.2.2.1

Multiply $f$ by $f^{2}$ .

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Step 1.5.2.2.1.1

Raise $f$ to the power of $1$ .

$f^{1} f^{2} + f \cdot - 1 - 1 f$

Step 1.5.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{1 + 2} + f \cdot - 1 - 1 f$

Step 1.5.2.2.2

Add $1$ and $2$ .

$f^{3} + f \cdot - 1 - 1 f$

Step 1.5.2.3

Move $- 1$ to the left of $f$ .

$f^{3} - 1 \cdot f - 1 f$

Step 1.5.2.4

Rewrite $- 1 f$ as $- f$ .

$f^{3} - f - 1 f$

Step 1.5.2.5

Rewrite $- 1 f$ as $- f$ .

$f^{3} - f - f$

Step 1.5.3

Subtract $f$ from $- f$ .

$f^{3} - 2 f$

Step 2

Since the determinant is non-zero, the inverse exists.

Step 3

Set up a $3 \times 6$ matrix where the left half is the original matrix and the right half is its identity matrix.

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

Step 4

Find the reduced row echelon form.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Simplify $R_{1}$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify $R_{2}$ .

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

Simplify $R_{2}$ .

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Simplify $R_{3}$ .

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

Step 4.5

Multiply each element of $R_{3}$ by $\frac{1}{f - \frac{f}{(f + 1) (f - 1)}}$ to make the entry at $3, 3$ a $1$ .

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Step 4.5.1

Multiply each element of $R_{3}$ by $\frac{1}{f - \frac{f}{(f + 1) (f - 1)}}$ to make the entry at $3, 3$ a $1$ .

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

Step 4.5.2

Simplify $R_{3}$ .

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

Step 4.6

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

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Step 4.6.1

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

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

Step 4.6.2

Simplify $R_{2}$ .

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

Simplify $R_{1}$ .

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

Step 5

The right half of the reduced row echelon form is the inverse.

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