Finite Math Examples

$[\begin{array}{c} 3 & 7 \\ 4 & 6 \end{array}] \cdot 9$

Step 1

Move $9$ to the left of $[\begin{array}{c} 3 & 7 \\ 4 & 6 \end{array}]$ .

$9 \cdot [\begin{array}{c} 3 & 7 \\ 4 & 6 \end{array}]$

Step 2

Multiply $9$ by each element of the matrix.

$[\begin{array}{c} 9 \cdot 3 & 9 \cdot 7 \\ 9 \cdot 4 & 9 \cdot 6 \end{array}]$

Step 3

Simplify each element in the matrix.

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Step 3.1

Multiply $9$ by $3$ .

$[\begin{array}{c} 27 & 9 \cdot 7 \\ 9 \cdot 4 & 9 \cdot 6 \end{array}]$

Step 3.2

Multiply $9$ by $7$ .

$[\begin{array}{c} 27 & 63 \\ 9 \cdot 4 & 9 \cdot 6 \end{array}]$

Step 3.3

Multiply $9$ by $4$ .

$[\begin{array}{c} 27 & 63 \\ 36 & 9 \cdot 6 \end{array}]$

Step 3.4

Multiply $9$ by $6$ .

$[\begin{array}{c} 27 & 63 \\ 36 & 54 \end{array}]$

Step 4

The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $a d - b c$ is the determinant.

Step 5

Find the determinant.

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Step 5.1

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

$27 \cdot 54 - 36 \cdot 63$

Step 5.2

Simplify the determinant.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $27$ by $54$ .

$1458 - 36 \cdot 63$

Step 5.2.1.2

Multiply $- 36$ by $63$ .

$1458 - 2268$

Step 5.2.2

Subtract $2268$ from $1458$ .

$- 810$

Step 6

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

Step 7

Substitute the known values into the formula for the inverse.

$\frac{1}{- 810} [\begin{array}{c} 54 & - 63 \\ - 36 & 27 \end{array}]$

Step 8

Move the negative in front of the fraction.

$- \frac{1}{810} [\begin{array}{c} 54 & - 63 \\ - 36 & 27 \end{array}]$

Step 9

Multiply $- \frac{1}{810}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{810} \cdot 54 & - \frac{1}{810} \cdot - 63 \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10

Simplify each element in the matrix.

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Step 10.1

Cancel the common factor of $54$ .

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Step 10.1.1

Move the leading negative in $- \frac{1}{810}$ into the numerator.

$[\begin{array}{c} \frac{- 1}{810} \cdot 54 & - \frac{1}{810} \cdot - 63 \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.1.2

Factor $54$ out of $810$ .

$[\begin{array}{c} \frac{- 1}{54 (15)} \cdot 54 & - \frac{1}{810} \cdot - 63 \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.1.3

Cancel the common factor.

$[\begin{array}{c} \frac{- 1}{54 \cdot 15} \cdot 54 & - \frac{1}{810} \cdot - 63 \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.1.4

Rewrite the expression.

$[\begin{array}{c} \frac{- 1}{15} & - \frac{1}{810} \cdot - 63 \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.2

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{15} & - \frac{1}{810} \cdot - 63 \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.3

Cancel the common factor of $9$ .

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Step 10.3.1

Move the leading negative in $- \frac{1}{810}$ into the numerator.

$[\begin{array}{c} - \frac{1}{15} & \frac{- 1}{810} \cdot - 63 \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.3.2

Factor $9$ out of $810$ .

$[\begin{array}{c} - \frac{1}{15} & \frac{- 1}{9 (90)} \cdot - 63 \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.3.3

Factor $9$ out of $- 63$ .

$[\begin{array}{c} - \frac{1}{15} & \frac{- 1}{9 \cdot 90} \cdot (9 \cdot - 7) \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.3.4

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{15} & \frac{- 1}{9 \cdot 90} \cdot (9 \cdot - 7) \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.3.5

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{15} & \frac{- 1}{90} \cdot - 7 \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.4

Combine $\frac{- 1}{90}$ and $- 7$ .

$[\begin{array}{c} - \frac{1}{15} & \frac{- - 7}{90} \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.5

Multiply $- 1$ by $- 7$ .

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ - \frac{1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.6

Cancel the common factor of $18$ .

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Step 10.6.1

Move the leading negative in $- \frac{1}{810}$ into the numerator.

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{- 1}{810} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.6.2

Factor $18$ out of $810$ .

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{- 1}{18 (45)} \cdot - 36 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.6.3

Factor $18$ out of $- 36$ .

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{- 1}{18 \cdot 45} \cdot (18 \cdot - 2) & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.6.4

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{- 1}{18 \cdot 45} \cdot (18 \cdot - 2) & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.6.5

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{- 1}{45} \cdot - 2 & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.7

Combine $\frac{- 1}{45}$ and $- 2$ .

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{- - 2}{45} & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.8

Multiply $- 1$ by $- 2$ .

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{2}{45} & - \frac{1}{810} \cdot 27 \end{array}]$

Step 10.9

Cancel the common factor of $27$ .

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Step 10.9.1

Move the leading negative in $- \frac{1}{810}$ into the numerator.

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{2}{45} & \frac{- 1}{810} \cdot 27 \end{array}]$

Step 10.9.2

Factor $27$ out of $810$ .

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{2}{45} & \frac{- 1}{27 (30)} \cdot 27 \end{array}]$

Step 10.9.3

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{2}{45} & \frac{- 1}{27 \cdot 30} \cdot 27 \end{array}]$

Step 10.9.4

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{2}{45} & \frac{- 1}{30} \end{array}]$

Step 10.10

Move the negative in front of the fraction.

$[\begin{array}{c} - \frac{1}{15} & \frac{7}{90} \\ \frac{2}{45} & - \frac{1}{30} \end{array}]$