Finite Math Examples

${[\begin{array}{c} 5 & - 7 \\ 4 & 11 \end{array}]}^{2}$

Step 1

To evaluate a square matrix to a positive integer power $n$ , multiply $n$ copies of the matrix.

$[\begin{array}{c} 5 & - 7 \\ 4 & 11 \end{array}] \cdot [\begin{array}{c} 5 & - 7 \\ 4 & 11 \end{array}]$

Step 2

Multiply $[\begin{array}{c} 5 & - 7 \\ 4 & 11 \end{array}] [\begin{array}{c} 5 & - 7 \\ 4 & 11 \end{array}]$ .

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Step 2.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 2$ .

Step 2.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 5 \cdot 5 - 7 \cdot 4 & 5 \cdot - 7 - 7 \cdot 11 \\ 4 \cdot 5 + 11 \cdot 4 & 4 \cdot - 7 + 11 \cdot 11 \end{array}]$

Step 2.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} - 3 & - 112 \\ 64 & 93 \end{array}]$

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

$- 3 \cdot 93 - 64 \cdot - 112$

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

Multiply $- 3$ by $93$ .

$- 279 - 64 \cdot - 112$

Step 4.2.1.2

Multiply $- 64$ by $- 112$ .

$- 279 + 7168$

Step 4.2.2

Add $- 279$ and $7168$ .

$6889$

Step 5

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

Step 6

Substitute the known values into the formula for the inverse.

$\frac{1}{6889} [\begin{array}{c} 93 & 112 \\ - 64 & - 3 \end{array}]$

Step 7

Multiply $\frac{1}{6889}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{6889} \cdot 93 & \frac{1}{6889} \cdot 112 \\ \frac{1}{6889} \cdot - 64 & \frac{1}{6889} \cdot - 3 \end{array}]$

Step 8

Simplify each element in the matrix.

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Step 8.1

Combine $\frac{1}{6889}$ and $93$ .

$[\begin{array}{c} \frac{93}{6889} & \frac{1}{6889} \cdot 112 \\ \frac{1}{6889} \cdot - 64 & \frac{1}{6889} \cdot - 3 \end{array}]$

Step 8.2

Combine $\frac{1}{6889}$ and $112$ .

$[\begin{array}{c} \frac{93}{6889} & \frac{112}{6889} \\ \frac{1}{6889} \cdot - 64 & \frac{1}{6889} \cdot - 3 \end{array}]$

Step 8.3

Combine $\frac{1}{6889}$ and $- 64$ .

$[\begin{array}{c} \frac{93}{6889} & \frac{112}{6889} \\ \frac{- 64}{6889} & \frac{1}{6889} \cdot - 3 \end{array}]$

Step 8.4

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{93}{6889} & \frac{112}{6889} \\ - \frac{64}{6889} & \frac{1}{6889} \cdot - 3 \end{array}]$

Step 8.5

Combine $\frac{1}{6889}$ and $- 3$ .

$[\begin{array}{c} \frac{93}{6889} & \frac{112}{6889} \\ - \frac{64}{6889} & \frac{- 3}{6889} \end{array}]$

Step 8.6

Move the negative in front of the fraction.

$[\begin{array}{c} \frac{93}{6889} & \frac{112}{6889} \\ - \frac{64}{6889} & - \frac{3}{6889} \end{array}]$