Linear Algebra Examples

$[\begin{array}{c} 32 & 10 \\ \frac{3}{5} & \frac{1}{8} \end{array}] \cdot F = [\begin{array}{c} - 80 & 80 \\ 1 & 2 \end{array}]$

Step 1

Find the inverse of $[\begin{array}{c} 32 & 10 \\ \frac{3}{5} & \frac{1}{8} \end{array}]$ .

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Step 1.1

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 1.2

Find the determinant.

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Step 1.2.1

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

$32 (\frac{1}{8}) - \frac{3}{5} \cdot 10$

Step 1.2.2

Simplify the determinant.

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Cancel the common factor of $8$ .

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Step 1.2.2.1.1.1

Factor $8$ out of $32$ .

$8 (4) \frac{1}{8} - \frac{3}{5} \cdot 10$

Step 1.2.2.1.1.2

Cancel the common factor.

$8 \cdot 4 \frac{1}{8} - \frac{3}{5} \cdot 10$

Step 1.2.2.1.1.3

Rewrite the expression.

$4 - \frac{3}{5} \cdot 10$

Step 1.2.2.1.2

Cancel the common factor of $5$ .

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Step 1.2.2.1.2.1

Move the leading negative in $- \frac{3}{5}$ into the numerator.

$4 + \frac{- 3}{5} \cdot 10$

Step 1.2.2.1.2.2

Factor $5$ out of $10$ .

$4 + \frac{- 3}{5} \cdot (5 (2))$

Step 1.2.2.1.2.3

Cancel the common factor.

$4 + \frac{- 3}{5} \cdot (5 \cdot 2)$

Step 1.2.2.1.2.4

Rewrite the expression.

$4 - 3 \cdot 2$

Step 1.2.2.1.3

Multiply $- 3$ by $2$ .

$4 - 6$

Step 1.2.2.2

Subtract $6$ from $4$ .

$- 2$

Step 1.3

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

Step 1.4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 2} [\begin{array}{c} \frac{1}{8} & - 10 \\ - \frac{3}{5} & 32 \end{array}]$

Step 1.5

Move the negative in front of the fraction.

$- \frac{1}{2} [\begin{array}{c} \frac{1}{8} & - 10 \\ - \frac{3}{5} & 32 \end{array}]$

Step 1.6

Multiply $- \frac{1}{2}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{2} \cdot \frac{1}{8} & - \frac{1}{2} \cdot - 10 \\ - \frac{1}{2} (- \frac{3}{5}) & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7

Simplify each element in the matrix.

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Step 1.7.1

Multiply $- \frac{1}{2} \cdot \frac{1}{8}$ .

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Step 1.7.1.1

Multiply $\frac{1}{8}$ by $\frac{1}{2}$ .

$[\begin{array}{c} - \frac{1}{8 \cdot 2} & - \frac{1}{2} \cdot - 10 \\ - \frac{1}{2} (- \frac{3}{5}) & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.1.2

Multiply $8$ by $2$ .

$[\begin{array}{c} - \frac{1}{16} & - \frac{1}{2} \cdot - 10 \\ - \frac{1}{2} (- \frac{3}{5}) & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.2

Cancel the common factor of $2$ .

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Step 1.7.2.1

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

$[\begin{array}{c} - \frac{1}{16} & \frac{- 1}{2} \cdot - 10 \\ - \frac{1}{2} (- \frac{3}{5}) & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.2.2

Factor $2$ out of $- 10$ .

$[\begin{array}{c} - \frac{1}{16} & \frac{- 1}{2} \cdot (2 (- 5)) \\ - \frac{1}{2} (- \frac{3}{5}) & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.2.3

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{16} & \frac{- 1}{2} \cdot (2 \cdot - 5) \\ - \frac{1}{2} (- \frac{3}{5}) & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.2.4

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{16} & - 1 \cdot - 5 \\ - \frac{1}{2} (- \frac{3}{5}) & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.3

Multiply $- 1$ by $- 5$ .

$[\begin{array}{c} - \frac{1}{16} & 5 \\ - \frac{1}{2} (- \frac{3}{5}) & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.4

Multiply $- \frac{1}{2} (- \frac{3}{5})$ .

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Step 1.7.4.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{1}{16} & 5 \\ 1 (\frac{1}{2}) \frac{3}{5} & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.4.2

Multiply $\frac{1}{2}$ by $1$ .

$[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{1}{2} \cdot \frac{3}{5} & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.4.3

Multiply $\frac{1}{2}$ by $\frac{3}{5}$ .

$[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{2 \cdot 5} & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.4.4

Multiply $2$ by $5$ .

$[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - \frac{1}{2} \cdot 32 \end{array}]$

Step 1.7.5

Cancel the common factor of $2$ .

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Step 1.7.5.1

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

$[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & \frac{- 1}{2} \cdot 32 \end{array}]$

Step 1.7.5.2

Factor $2$ out of $32$ .

$[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & \frac{- 1}{2} \cdot (2 (16)) \end{array}]$

Step 1.7.5.3

Cancel the common factor.

$[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & \frac{- 1}{2} \cdot (2 \cdot 16) \end{array}]$

Step 1.7.5.4

Rewrite the expression.

$[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - 1 \cdot 16 \end{array}]$

Step 1.7.6

Multiply $- 1$ by $16$ .

$[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - 16 \end{array}]$

Step 2

Multiply both sides by the inverse of $[\begin{array}{c} 32 & 10 \\ \frac{3}{5} & \frac{1}{8} \end{array}]$ .

$[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - 16 \end{array}] [\begin{array}{c} 32 & 10 \\ \frac{3}{5} & \frac{1}{8} \end{array}] F = [\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - 16 \end{array}] [\begin{array}{c} - 80 & 80 \\ 1 & 2 \end{array}]$

Step 3

Simplify the equation.

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Step 3.1

Multiply $[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - 16 \end{array}] [\begin{array}{c} 32 & 10 \\ \frac{3}{5} & \frac{1}{8} \end{array}]$ .

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Step 3.1.1

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 3.1.2

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

$[\begin{array}{c} - \frac{1}{16} \cdot 32 + 5 (\frac{3}{5}) & - \frac{1}{16} \cdot 10 + 5 (\frac{1}{8}) \\ \frac{3}{10} \cdot 32 - 16 (\frac{3}{5}) & \frac{3}{10} \cdot 10 - 16 (\frac{1}{8}) \end{array}] F = [\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - 16 \end{array}] [\begin{array}{c} - 80 & 80 \\ 1 & 2 \end{array}]$

Step 3.1.3

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] F = [\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - 16 \end{array}] [\begin{array}{c} - 80 & 80 \\ 1 & 2 \end{array}]$

Step 3.2

Multiplying the identity matrix by any matrix $A$ is the matrix $A$ itself.

$F = [\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - 16 \end{array}] [\begin{array}{c} - 80 & 80 \\ 1 & 2 \end{array}]$

Step 3.3

Multiply $[\begin{array}{c} - \frac{1}{16} & 5 \\ \frac{3}{10} & - 16 \end{array}] [\begin{array}{c} - 80 & 80 \\ 1 & 2 \end{array}]$ .

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Step 3.3.1

Step 3.3.2

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

$F = [\begin{array}{c} - \frac{1}{16} \cdot - 80 + 5 \cdot 1 & - \frac{1}{16} \cdot 80 + 5 \cdot 2 \\ \frac{3}{10} \cdot - 80 - 16 \cdot 1 & \frac{3}{10} \cdot 80 - 16 \cdot 2 \end{array}]$

Step 3.3.3

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

$F = [\begin{array}{c} 10 & 5 \\ - 40 & - 8 \end{array}]$