Finite Math Examples

$M [\begin{array}{c} 3 & 1 \\ 1 & 5 \end{array}]$

Step 1

Multiply $M$ by each element of the matrix.

$[\begin{array}{c} M \cdot 3 & M \cdot 1 \\ M \cdot 1 & M \cdot 5 \end{array}]$

Step 2

Simplify each element in the matrix.

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Step 2.1

Move $3$ to the left of $M$ .

$[\begin{array}{c} 3 M & M \cdot 1 \\ M \cdot 1 & M \cdot 5 \end{array}]$

Step 2.2

Multiply $M$ by $1$ .

$[\begin{array}{c} 3 M & M \\ M \cdot 1 & M \cdot 5 \end{array}]$

Step 2.3

Multiply $M$ by $1$ .

$[\begin{array}{c} 3 M & M \\ M & M \cdot 5 \end{array}]$

Step 2.4

Move $5$ to the left of $M$ .

$[\begin{array}{c} 3 M & M \\ M & 5 M \end{array}]$

Step 3

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 4

Find the determinant.

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Step 4.1

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

$3 M (5 M) - M \cdot M$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 5 M \cdot M - M \cdot M$

Step 4.2.1.2

Multiply $M$ by $M$ by adding the exponents.

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Step 4.2.1.2.1

Move $M$ .

$3 \cdot 5 (M \cdot M) - M \cdot M$

Step 4.2.1.2.2

Multiply $M$ by $M$ .

$3 \cdot 5 M^{2} - M \cdot M$

Step 4.2.1.3

Multiply $3$ by $5$ .

$15 M^{2} - M \cdot M$

Step 4.2.1.4

Multiply $M$ by $M$ by adding the exponents.

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Step 4.2.1.4.1

Move $M$ .

$15 M^{2} - (M \cdot M)$

Step 4.2.1.4.2

Multiply $M$ by $M$ .

$15 M^{2} - M^{2}$

Step 4.2.2

Subtract $M^{2}$ from $15 M^{2}$ .

$14 M^{2}$

Step 5

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

Step 6

Substitute the known values into the formula for the inverse.

$\frac{1}{14 M^{2}} [\begin{array}{c} 5 M & - M \\ - M & 3 M \end{array}]$

Step 7

Multiply $\frac{1}{14 M^{2}}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{14 M^{2}} (5 M) & \frac{1}{14 M^{2}} (- M) \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8

Simplify each element in the matrix.

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Step 8.1

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} 5 \frac{1}{14 M^{2}} M & \frac{1}{14 M^{2}} (- M) \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.2

Combine $5$ and $\frac{1}{14 M^{2}}$ .

$[\begin{array}{c} \frac{5}{14 M^{2}} M & \frac{1}{14 M^{2}} (- M) \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.3

Cancel the common factor of $M$ .

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Step 8.3.1

Factor $M$ out of $14 M^{2}$ .

$[\begin{array}{c} \frac{5}{M (14 M)} M & \frac{1}{14 M^{2}} (- M) \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.3.2

Cancel the common factor.

$[\begin{array}{c} \frac{5}{M (14 M)} M & \frac{1}{14 M^{2}} (- M) \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.3.3

Rewrite the expression.

$[\begin{array}{c} \frac{5}{14 M} & \frac{1}{14 M^{2}} (- M) \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.4

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M^{2}} M \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.5

Cancel the common factor of $M$ .

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Step 8.5.1

Move the leading negative in $- \frac{1}{14 M^{2}}$ into the numerator.

$[\begin{array}{c} \frac{5}{14 M} & \frac{- 1}{14 M^{2}} M \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.5.2

Factor $M$ out of $14 M^{2}$ .

$[\begin{array}{c} \frac{5}{14 M} & \frac{- 1}{M (14 M)} M \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.5.3

Cancel the common factor.

$[\begin{array}{c} \frac{5}{14 M} & \frac{- 1}{M (14 M)} M \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.5.4

Rewrite the expression.

$[\begin{array}{c} \frac{5}{14 M} & \frac{- 1}{14 M} \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.6

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ \frac{1}{14 M^{2}} (- M) & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.7

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ - \frac{1}{14 M^{2}} M & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.8

Cancel the common factor of $M$ .

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Step 8.8.1

Move the leading negative in $- \frac{1}{14 M^{2}}$ into the numerator.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ \frac{- 1}{14 M^{2}} M & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.8.2

Factor $M$ out of $14 M^{2}$ .

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ \frac{- 1}{M (14 M)} M & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.8.3

Cancel the common factor.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ \frac{- 1}{M (14 M)} M & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.8.4

Rewrite the expression.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ \frac{- 1}{14 M} & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.9

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ - \frac{1}{14 M} & \frac{1}{14 M^{2}} (3 M) \end{array}]$

Step 8.10

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ - \frac{1}{14 M} & 3 \frac{1}{14 M^{2}} M \end{array}]$

Step 8.11

Combine $3$ and $\frac{1}{14 M^{2}}$ .

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ - \frac{1}{14 M} & \frac{3}{14 M^{2}} M \end{array}]$

Step 8.12

Cancel the common factor of $M$ .

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Step 8.12.1

Factor $M$ out of $14 M^{2}$ .

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ - \frac{1}{14 M} & \frac{3}{M (14 M)} M \end{array}]$

Step 8.12.2

Cancel the common factor.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ - \frac{1}{14 M} & \frac{3}{M (14 M)} M \end{array}]$

Step 8.12.3

Rewrite the expression.

$[\begin{array}{c} \frac{5}{14 M} & - \frac{1}{14 M} \\ - \frac{1}{14 M} & \frac{3}{14 M} \end{array}]$