Finite Math Examples

[\begin{matrix} 5 & 9 2 & 3 \end{matrix}] + 6

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

Step 1

Adding

6

$6$ to a

2 \times 2

$2 \times 2$ square matrix is the same as adding

6

$6$ times the

2 \times 2

$2 \times 2$ identity matrix.

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

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

Step 2

Multiply

6

$6$ by each element of the matrix.

[\begin{matrix} 5 & 9 2 & 3 \end{matrix}] + [\begin{matrix} 6 \cdot 1 & 6 \cdot 0 6 \cdot 0 & 6 \cdot 1 \end{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.

Tap for more steps...

Step 3.1

Multiply

6

$6$ by

1

$1$ .

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

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

Step 3.2

Multiply

6

$6$ by

0

$0$ .

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

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

Step 3.3

Multiply

6

$6$ by

0

$0$ .

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

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

Step 3.4

Multiply

6

$6$ by

1

$1$ .

[\begin{matrix} 5 & 9 2 & 3 \end{matrix}] + [\begin{matrix} 6 & 0 0 & 6 \end{matrix}]

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

[\begin{matrix} 5 & 9 2 & 3 \end{matrix}] + [\begin{matrix} 6 & 0 0 & 6 \end{matrix}]

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

Step 4

Add the corresponding elements.

[\begin{matrix} 5 + 6 & 9 + 0 2 + 0 & 3 + 6 \end{matrix}]

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

Step 5

Simplify each element.

Tap for more steps...

Step 5.1

Add

5

$5$ and

6

$6$ .

[\begin{matrix} 11 & 9 + 0 2 + 0 & 3 + 6 \end{matrix}]

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

Step 5.2

Add

9

$9$ and

0

$0$ .

[\begin{matrix} 11 & 9 2 + 0 & 3 + 6 \end{matrix}]

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

Step 5.3

Add

2

$2$ and

0

$0$ .

[\begin{matrix} 11 & 9 2 & 3 + 6 \end{matrix}]

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

Step 5.4

Add

3

$3$ and

6

$6$ .

[\begin{matrix} 11 & 9 2 & 9 \end{matrix}]

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

[\begin{matrix} 11 & 9 2 & 9 \end{matrix}]

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

Step 6

The inverse of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

\frac{1}{a d - b c} [\begin{matrix} d & - b - c & a \end{matrix}]

$\frac{1}{a d - b c} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where

a d - b c

$a d - b c$ is the determinant.

Step 7

Find the determinant.

Tap for more steps...

Step 7.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

$| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

11 \cdot 9 - 2 \cdot 9

$11 \cdot 9 - 2 \cdot 9$

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

11

$11$ by

9

$9$ .

99 - 2 \cdot 9

$99 - 2 \cdot 9$

Step 7.2.1.2

Multiply

- 2

$- 2$ by

9

$9$ .

99 - 18

$99 - 18$

99 - 18

$99 - 18$

Step 7.2.2

Subtract

18

$18$ from

99

$99$ .

81

$81$

81

$81$

81

$81$

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{matrix} 9 & - 9 - 2 & 11 \end{matrix}]

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

Step 10

Multiply

\frac{1}{81}

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

[\begin{matrix} \frac{1}{81} \cdot 9 & \frac{1}{81} \cdot - 9 \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{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.

Tap for more steps...

Step 11.1

Cancel the common factor of

9

$9$ .

Tap for more steps...

Step 11.1.1

Factor

9

$9$ out of

81

$81$ .

⎡ ⎣ \begin{matrix} \frac{1}{9 (9)} \cdot 9 & \frac{1}{81} \cdot - 9 \frac{1}{81} \cdot - 2 & \frac{1}{81} \cdot 11 \end{matrix} ⎤ ⎦

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

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

$9$ .

Tap for more steps...

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

$11$ .

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