Finite Math Examples

$[\begin{array}{c} 5 & 9 \\ 2 & 3 \end{array}] + 6$

Step 1

Adding $6$ to a $2 \times 2$ square matrix is the same as adding $6$ times the $2 \times 2$ identity matrix.

$[\begin{array}{c} 5 & 9 \\ 2 & 3 \end{array}] + 6 [\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}]$

Step 2

Multiply $6$ by each element of the matrix.

$[\begin{array}{c} 5 & 9 \\ 2 & 3 \end{array}] + [\begin{array}{c} 6 \cdot 1 & 6 \cdot 0 \\ 6 \cdot 0 & 6 \cdot 1 \end{array}]$

Step 3

Simplify each element in the matrix.

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Step 3.1

Multiply $6$ by $1$ .

$[\begin{array}{c} 5 & 9 \\ 2 & 3 \end{array}] + [\begin{array}{c} 6 & 6 \cdot 0 \\ 6 \cdot 0 & 6 \cdot 1 \end{array}]$

Step 3.2

Multiply $6$ by $0$ .

$[\begin{array}{c} 5 & 9 \\ 2 & 3 \end{array}] + [\begin{array}{c} 6 & 0 \\ 6 \cdot 0 & 6 \cdot 1 \end{array}]$

Step 3.3

Multiply $6$ by $0$ .

$[\begin{array}{c} 5 & 9 \\ 2 & 3 \end{array}] + [\begin{array}{c} 6 & 0 \\ 0 & 6 \cdot 1 \end{array}]$

Step 3.4

Multiply $6$ by $1$ .

$[\begin{array}{c} 5 & 9 \\ 2 & 3 \end{array}] + [\begin{array}{c} 6 & 0 \\ 0 & 6 \end{array}]$

Step 4

Add the corresponding elements.

$[\begin{array}{c} 5 + 6 & 9 + 0 \\ 2 + 0 & 3 + 6 \end{array}]$

Step 5

Simplify each element.

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Step 5.1

Add $5$ and $6$ .

$[\begin{array}{c} 11 & 9 + 0 \\ 2 + 0 & 3 + 6 \end{array}]$

Step 5.2

Add $9$ and $0$ .

$[\begin{array}{c} 11 & 9 \\ 2 + 0 & 3 + 6 \end{array}]$

Step 5.3

Add $2$ and $0$ .

$[\begin{array}{c} 11 & 9 \\ 2 & 3 + 6 \end{array}]$

Step 5.4

Add $3$ and $6$ .

$[\begin{array}{c} 11 & 9 \\ 2 & 9 \end{array}]$

Step 6

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 7

Find the determinant.

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Step 7.1

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

$11 \cdot 9 - 2 \cdot 9$

Step 7.2

Simplify the determinant.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $11$ by $9$ .

$99 - 2 \cdot 9$

Step 7.2.1.2

Multiply $- 2$ by $9$ .

$99 - 18$

Step 7.2.2

Subtract $18$ from $99$ .

$81$

Step 8

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

Step 9

Substitute the known values into the formula for the inverse.

$\frac{1}{81} [\begin{array}{c} 9 & - 9 \\ - 2 & 11 \end{array}]$

Step 10

Multiply $\frac{1}{81}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{81} \cdot 9 & \frac{1}{81} \cdot - 9 \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11

Simplify each element in the matrix.

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Step 11.1

Cancel the common factor of $9$ .

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Step 11.1.1

Factor $9$ out of $81$ .

$[\begin{array}{c} \frac{1}{9 (9)} \cdot 9 & \frac{1}{81} \cdot - 9 \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.1.2

Cancel the common factor.

$[\begin{array}{c} \frac{1}{9 \cdot 9} \cdot 9 & \frac{1}{81} \cdot - 9 \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.1.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{9} & \frac{1}{81} \cdot - 9 \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.2

Cancel the common factor of $9$ .

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Step 11.2.1

Factor $9$ out of $81$ .

$[\begin{array}{c} \frac{1}{9} & \frac{1}{9 (9)} \cdot - 9 \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.2.2

Factor $9$ out of $- 9$ .

$[\begin{array}{c} \frac{1}{9} & \frac{1}{9 \cdot 9} \cdot (9 \cdot - 1) \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.2.3

Cancel the common factor.

$[\begin{array}{c} \frac{1}{9} & \frac{1}{9 \cdot 9} \cdot (9 \cdot - 1) \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.2.4

Rewrite the expression.

$[\begin{array}{c} \frac{1}{9} & \frac{1}{9} \cdot - 1 \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.3

Combine $\frac{1}{9}$ and $- 1$ .

$[\begin{array}{c} \frac{1}{9} & \frac{- 1}{9} \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.4

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{9} & - \frac{1}{9} \\ \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.5

Combine $\frac{1}{81}$ and $- 2$ .

$[\begin{array}{c} \frac{1}{9} & - \frac{1}{9} \\ \frac{- 2}{81} & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.6

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{1}{9} & - \frac{1}{9} \\ - \frac{2}{81} & \frac{1}{81} \cdot 11 \end{array}]$

Step 11.7

Combine $\frac{1}{81}$ and $11$ .

$[\begin{array}{c} \frac{1}{9} & - \frac{1}{9} \\ - \frac{2}{81} & \frac{11}{81} \end{array}]$