Finite Math Examples

$[\begin{array}{c} - 1 & 0 & 1 \\ 0.3 & - 1 & 0 \\ 0.7 & 1 & - 1 \end{array}]$

Step 1

Find the determinant.

Tap for more steps...

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.

Tap for more steps...

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.

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

Step 1.1.4

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

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

Step 1.1.5

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

$| \begin{array}{c} 0.3 & 0 \\ 0.7 & - 1 \end{array} |$

Step 1.1.6

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

$0 | \begin{array}{c} 0.3 & 0 \\ 0.7 & - 1 \end{array} |$

Step 1.1.7

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

$| \begin{array}{c} 0.3 & - 1 \\ 0.7 & 1 \end{array} |$

Step 1.1.8

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

$1 | \begin{array}{c} 0.3 & - 1 \\ 0.7 & 1 \end{array} |$

Step 1.1.9

Add the terms together.

$- 1 | \begin{array}{c} - 1 & 0 \\ 1 & - 1 \end{array} | + 0 | \begin{array}{c} 0.3 & 0 \\ 0.7 & - 1 \end{array} | + 1 | \begin{array}{c} 0.3 & - 1 \\ 0.7 & 1 \end{array} |$

Step 1.2

Multiply $0$ by $| \begin{array}{c} 0.3 & 0 \\ 0.7 & - 1 \end{array} |$ .

$- 1 | \begin{array}{c} - 1 & 0 \\ 1 & - 1 \end{array} | + 0 + 1 | \begin{array}{c} 0.3 & - 1 \\ 0.7 & 1 \end{array} |$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$- 1 (- - 1 - 1 \cdot 0) + 0 + 1 | \begin{array}{c} 0.3 & - 1 \\ 0.7 & 1 \end{array} |$

Step 1.3.2

Simplify the determinant.

Tap for more steps...

Step 1.3.2.1

Multiply $- 1$ by $- 1$ .

$- 1 (1 - 1 \cdot 0) + 0 + 1 | \begin{array}{c} 0.3 & - 1 \\ 0.7 & 1 \end{array} |$

Step 1.3.2.2

Subtract $0$ from $1$ .

$- 1 \cdot 1 + 0 + 1 | \begin{array}{c} 0.3 & - 1 \\ 0.7 & 1 \end{array} |$

Step 1.4

Evaluate $| \begin{array}{c} 0.3 & - 1 \\ 0.7 & 1 \end{array} |$ .

Tap for more steps...

Step 1.4.1

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

$- 1 \cdot 1 + 0 + 1 (0.3 \cdot 1 - 0.7 \cdot - 1)$

Step 1.4.2

Simplify the determinant.

Tap for more steps...

Step 1.4.2.1

Simplify each term.

Tap for more steps...

Step 1.4.2.1.1

Multiply $0.3$ by $1$ .

$- 1 \cdot 1 + 0 + 1 (0.3 - 0.7 \cdot - 1)$

Step 1.4.2.1.2

Multiply $- 0.7$ by $- 1$ .

$- 1 \cdot 1 + 0 + 1 (0.3 + 0.7)$

Step 1.4.2.2

Add $0.3$ and $0.7$ .

$- 1 \cdot 1 + 0 + 1 \cdot 1$

Step 1.5

Simplify the determinant.

Tap for more steps...

Step 1.5.1

Simplify each term.

Tap for more steps...

Step 1.5.1.1

Multiply $- 1$ by $1$ .

$- 1 + 0 + 1 \cdot 1$

Step 1.5.1.2

Multiply $1$ by $1$ .

$- 1 + 0 + 1$

Step 1.5.2

Add $- 1$ and $0$ .

$- 1 + 1$

Step 1.5.3

Add $- 1$ and $1$ .

$0$

Step 2

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