Finite Math Examples

$[\begin{array}{c} - 0.12 & - 0.02 & 0.18 \\ 0 & 0.1 & - 0.3 \\ 0.06 & - 0.14 & 0.06 \end{array}] \cdot - 0.06$

Step 1

Move $- 0.06$ to the left of $[\begin{array}{c} - 0.12 & - 0.02 & 0.18 \\ 0 & 0.1 & - 0.3 \\ 0.06 & - 0.14 & 0.06 \end{array}]$ .

$- 0.06 \cdot [\begin{array}{c} - 0.12 & - 0.02 & 0.18 \\ 0 & 0.1 & - 0.3 \\ 0.06 & - 0.14 & 0.06 \end{array}]$

Step 2

Multiply $- 0.06$ by each element of the matrix.

$[\begin{array}{c} - 0.06 \cdot - 0.12 & - 0.06 \cdot - 0.02 & - 0.06 \cdot 0.18 \\ - 0.06 \cdot 0 & - 0.06 \cdot 0.1 & - 0.06 \cdot - 0.3 \\ - 0.06 \cdot 0.06 & - 0.06 \cdot - 0.14 & - 0.06 \cdot 0.06 \end{array}]$

Step 3

Simplify each element in the matrix.

Tap for more steps...

Step 3.1

Multiply $- 0.06$ by $- 0.12$ .

$[\begin{array}{c} 0.0072 & - 0.06 \cdot - 0.02 & - 0.06 \cdot 0.18 \\ - 0.06 \cdot 0 & - 0.06 \cdot 0.1 & - 0.06 \cdot - 0.3 \\ - 0.06 \cdot 0.06 & - 0.06 \cdot - 0.14 & - 0.06 \cdot 0.06 \end{array}]$

Step 3.2

Multiply $- 0.06$ by $- 0.02$ .

$[\begin{array}{c} 0.0072 & 0.0012 & - 0.06 \cdot 0.18 \\ - 0.06 \cdot 0 & - 0.06 \cdot 0.1 & - 0.06 \cdot - 0.3 \\ - 0.06 \cdot 0.06 & - 0.06 \cdot - 0.14 & - 0.06 \cdot 0.06 \end{array}]$

Step 3.3

Multiply $- 0.06$ by $0.18$ .

$[\begin{array}{c} 0.0072 & 0.0012 & - 0.0108 \\ - 0.06 \cdot 0 & - 0.06 \cdot 0.1 & - 0.06 \cdot - 0.3 \\ - 0.06 \cdot 0.06 & - 0.06 \cdot - 0.14 & - 0.06 \cdot 0.06 \end{array}]$

Step 3.4

Multiply $- 0.06$ by $0$ .

$[\begin{array}{c} 0.0072 & 0.0012 & - 0.0108 \\ 0 & - 0.06 \cdot 0.1 & - 0.06 \cdot - 0.3 \\ - 0.06 \cdot 0.06 & - 0.06 \cdot - 0.14 & - 0.06 \cdot 0.06 \end{array}]$

Step 3.5

Multiply $- 0.06$ by $0.1$ .

$[\begin{array}{c} 0.0072 & 0.0012 & - 0.0108 \\ 0 & - 0.006 & - 0.06 \cdot - 0.3 \\ - 0.06 \cdot 0.06 & - 0.06 \cdot - 0.14 & - 0.06 \cdot 0.06 \end{array}]$

Step 3.6

Multiply $- 0.06$ by $- 0.3$ .

$[\begin{array}{c} 0.0072 & 0.0012 & - 0.0108 \\ 0 & - 0.006 & 0.018 \\ - 0.06 \cdot 0.06 & - 0.06 \cdot - 0.14 & - 0.06 \cdot 0.06 \end{array}]$

Step 3.7

Multiply $- 0.06$ by $0.06$ .

$[\begin{array}{c} 0.0072 & 0.0012 & - 0.0108 \\ 0 & - 0.006 & 0.018 \\ - 0.0036 & - 0.06 \cdot - 0.14 & - 0.06 \cdot 0.06 \end{array}]$

Step 3.8

Multiply $- 0.06$ by $- 0.14$ .

$[\begin{array}{c} 0.0072 & 0.0012 & - 0.0108 \\ 0 & - 0.006 & 0.018 \\ - 0.0036 & 0.0084 & - 0.06 \cdot 0.06 \end{array}]$

Step 3.9

Multiply $- 0.06$ by $0.06$ .

$[\begin{array}{c} 0.0072 & 0.0012 & - 0.0108 \\ 0 & - 0.006 & 0.018 \\ - 0.0036 & 0.0084 & - 0.0036 \end{array}]$

Step 4

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 column $1$ by its cofactor and add.

Tap for more steps...

Step 4.1

Consider the corresponding sign chart.

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

Step 4.2

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

Step 4.3

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

$| \begin{array}{c} - 0.006 & 0.018 \\ 0.0084 & - 0.0036 \end{array} |$

Step 4.4

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

$0.0072 | \begin{array}{c} - 0.006 & 0.018 \\ 0.0084 & - 0.0036 \end{array} |$

Step 4.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

$| \begin{array}{c} 0.0012 & - 0.0108 \\ 0.0084 & - 0.0036 \end{array} |$

Step 4.6

Multiply element $a_{21}$ by its cofactor.

$0 | \begin{array}{c} 0.0012 & - 0.0108 \\ 0.0084 & - 0.0036 \end{array} |$

Step 4.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

$| \begin{array}{c} 0.0012 & - 0.0108 \\ - 0.006 & 0.018 \end{array} |$

Step 4.8

Multiply element $a_{31}$ by its cofactor.

$- 0.0036 | \begin{array}{c} 0.0012 & - 0.0108 \\ - 0.006 & 0.018 \end{array} |$

Step 4.9

Add the terms together.

$0.0072 | \begin{array}{c} - 0.006 & 0.018 \\ 0.0084 & - 0.0036 \end{array} | + 0 | \begin{array}{c} 0.0012 & - 0.0108 \\ 0.0084 & - 0.0036 \end{array} | - 0.0036 | \begin{array}{c} 0.0012 & - 0.0108 \\ - 0.006 & 0.018 \end{array} |$

Step 5

Multiply $0$ by $| \begin{array}{c} 0.0012 & - 0.0108 \\ 0.0084 & - 0.0036 \end{array} |$ .

$0.0072 | \begin{array}{c} - 0.006 & 0.018 \\ 0.0084 & - 0.0036 \end{array} | + 0 - 0.0036 | \begin{array}{c} 0.0012 & - 0.0108 \\ - 0.006 & 0.018 \end{array} |$

Step 6

Evaluate $| \begin{array}{c} - 0.006 & 0.018 \\ 0.0084 & - 0.0036 \end{array} |$ .

Tap for more steps...

Step 6.1

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

$0.0072 (- 0.006 \cdot - 0.0036 - 0.0084 \cdot 0.018) + 0 - 0.0036 | \begin{array}{c} 0.0012 & - 0.0108 \\ - 0.006 & 0.018 \end{array} |$

Step 6.2

Simplify the determinant.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Multiply $- 0.006$ by $- 0.0036$ .

$0.0072 (0.0000216 - 0.0084 \cdot 0.018) + 0 - 0.0036 | \begin{array}{c} 0.0012 & - 0.0108 \\ - 0.006 & 0.018 \end{array} |$

Step 6.2.1.2

Multiply $- 0.0084$ by $0.018$ .

$0.0072 (0.0000216 - 0.0001512) + 0 - 0.0036 | \begin{array}{c} 0.0012 & - 0.0108 \\ - 0.006 & 0.018 \end{array} |$

Step 6.2.2

Subtract $0.0001512$ from $0.0000216$ .

$0.0072 \cdot - 0.0001296 + 0 - 0.0036 | \begin{array}{c} 0.0012 & - 0.0108 \\ - 0.006 & 0.018 \end{array} |$

Step 7

Evaluate $| \begin{array}{c} 0.0012 & - 0.0108 \\ - 0.006 & 0.018 \end{array} |$ .

Tap for more steps...

Step 7.1

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

$0.0072 \cdot - 0.0001296 + 0 - 0.0036 (0.0012 \cdot 0.018 - (- 0.006 \cdot - 0.0108))$

Step 7.2

Simplify the determinant.

Tap for more steps...

Step 7.2.1

Simplify each term.

Tap for more steps...

Step 7.2.1.1

Multiply $0.0012$ by $0.018$ .

$0.0072 \cdot - 0.0001296 + 0 - 0.0036 (0.0000216 - (- 0.006 \cdot - 0.0108))$

Step 7.2.1.2

Multiply $- (- 0.006 \cdot - 0.0108)$ .

Tap for more steps...

Step 7.2.1.2.1

Multiply $- 0.006$ by $- 0.0108$ .

$0.0072 \cdot - 0.0001296 + 0 - 0.0036 (0.0000216 - 1 \cdot 0.0000648)$

Step 7.2.1.2.2

Multiply $- 1$ by $0.0000648$ .

$0.0072 \cdot - 0.0001296 + 0 - 0.0036 (0.0000216 - 0.0000648)$

Step 7.2.2

Subtract $0.0000648$ from $0.0000216$ .

$0.0072 \cdot - 0.0001296 + 0 - 0.0036 \cdot - 0.0000432$

Step 8

Simplify the determinant.

Tap for more steps...

Step 8.1

Simplify each term.

Tap for more steps...

Step 8.1.1

Multiply $0.0072$ by $- 0.0001296$ .

$- 0.00000093 + 0 - 0.0036 \cdot - 0.0000432$

Step 8.1.2

Multiply $- 0.0036$ by $- 0.0000432$ .

$- 0.00000093 + 0 + 0.00000015$

Step 8.2

Add $- 0.00000093$ and $0$ .

$- 0.00000093 + 0.00000015$

Step 8.3

Add $- 0.00000093$ and $0.00000015$ .

$- 0.000000\overline{7}$