Finite Math Examples

$[\begin{array}{c} 3 & 2 & 1 \\ 3 & 1 & 2 \\ 2 & 1 & 1 \end{array}]$

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.

$| \begin{array}{c} 1 & 2 \\ 1 & 1 \end{array} |$

Step 1.1.4

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

$3 | \begin{array}{c} 1 & 2 \\ 1 & 1 \end{array} |$

Step 1.1.5

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

$| \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$

Step 1.1.6

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

$- 2 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$

Step 1.1.7

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

$| \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.1.8

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

$1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.1.9

Add the terms together.

$3 | \begin{array}{c} 1 & 2 \\ 1 & 1 \end{array} | - 2 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.2

Evaluate $| \begin{array}{c} 1 & 2 \\ 1 & 1 \end{array} |$ .

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Step 1.2.1

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

$3 (1 \cdot 1 - 1 \cdot 2) - 2 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.2.2

Simplify the determinant.

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Multiply $1$ by $1$ .

$3 (1 - 1 \cdot 2) - 2 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.2.2.1.2

Multiply $- 1$ by $2$ .

$3 (1 - 2) - 2 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.2.2.2

Subtract $2$ from $1$ .

$3 \cdot - 1 - 2 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.3

Evaluate $| \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$ .

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Step 1.3.1

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

$3 \cdot - 1 - 2 (3 \cdot 1 - 2 \cdot 2) + 1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.3.2

Simplify the determinant.

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Multiply $3$ by $1$ .

$3 \cdot - 1 - 2 (3 - 2 \cdot 2) + 1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.3.2.1.2

Multiply $- 2$ by $2$ .

$3 \cdot - 1 - 2 (3 - 4) + 1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.3.2.2

Subtract $4$ from $3$ .

$3 \cdot - 1 - 2 \cdot - 1 + 1 | \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$

Step 1.4

Evaluate $| \begin{array}{c} 3 & 1 \\ 2 & 1 \end{array} |$ .

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Step 1.4.1

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

$3 \cdot - 1 - 2 \cdot - 1 + 1 (3 \cdot 1 - 2 \cdot 1)$

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 $3$ by $1$ .

$3 \cdot - 1 - 2 \cdot - 1 + 1 (3 - 2 \cdot 1)$

Step 1.4.2.1.2

Multiply $- 2$ by $1$ .

$3 \cdot - 1 - 2 \cdot - 1 + 1 (3 - 2)$

Step 1.4.2.2

Subtract $2$ from $3$ .

$3 \cdot - 1 - 2 \cdot - 1 + 1 \cdot 1$

Step 1.5

Simplify the determinant.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Multiply $3$ by $- 1$ .

$- 3 - 2 \cdot - 1 + 1 \cdot 1$

Step 1.5.1.2

Multiply $- 2$ by $- 1$ .

$- 3 + 2 + 1 \cdot 1$

Step 1.5.1.3

Multiply $1$ by $1$ .

$- 3 + 2 + 1$

Step 1.5.2

Add $- 3$ and $2$ .

$- 1 + 1$

Step 1.5.3

Add $- 1$ and $1$ .

$0$

Step 2

There is no inverse because the determinant is $0$ .