Linear Algebra Examples

$[\begin{array}{c} - e^{t} & 1 \\ e^{t} & e^{- t} \end{array}]$

Step 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 2

Find the determinant.

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Step 2.1

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

$- e^{t} e^{- t} - e^{t} \cdot 1$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $e^{t}$ by $e^{- t}$ by adding the exponents.

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Step 2.2.1.1

Move $e^{- t}$ .

$- (e^{- t} e^{t}) - e^{t} \cdot 1$

Step 2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- e^{- t + t} - e^{t} \cdot 1$

Step 2.2.1.3

Add $- t$ and $t$ .

$- e^{0} - e^{t} \cdot 1$

Step 2.2.2

Simplify $- e^{0}$ .

$- 1 - e^{t} \cdot 1$

Step 2.2.3

Multiply $- 1$ by $1$ .

$- 1 - e^{t}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{- 1 - e^{t}} [\begin{array}{c} e^{- t} & - 1 \\ - e^{t} & - e^{t} \end{array}]$

Step 5

Rewrite $- 1$ as $- 1 (1)$ .

$\frac{1}{- 1 (1) - e^{t}} [\begin{array}{c} e^{- t} & - 1 \\ - e^{t} & - e^{t} \end{array}]$

Step 6

Factor $- 1$ out of $- e^{t}$ .

$\frac{1}{- 1 (1) - (e^{t})} [\begin{array}{c} e^{- t} & - 1 \\ - e^{t} & - e^{t} \end{array}]$

Step 7

Factor $- 1$ out of $- 1 (1) - (e^{t})$ .

$\frac{1}{- 1 (1 + e^{t})} [\begin{array}{c} e^{- t} & - 1 \\ - e^{t} & - e^{t} \end{array}]$

Step 8

Move the negative in front of the fraction.

$- \frac{1}{1 + e^{t}} [\begin{array}{c} e^{- t} & - 1 \\ - e^{t} & - e^{t} \end{array}]$

Step 9

Multiply $- \frac{1}{1 + e^{t}}$ by each element of the matrix.

$[\begin{array}{c} - \frac{1}{1 + e^{t}} e^{- t} & - \frac{1}{1 + e^{t}} \cdot - 1 \\ - \frac{1}{1 + e^{t}} (- e^{t}) & - \frac{1}{1 + e^{t}} (- e^{t}) \end{array}]$

Step 10

Simplify each element in the matrix.

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Step 10.1

Combine $e^{- t}$ and $\frac{1}{1 + e^{t}}$ .

$[\begin{array}{c} - \frac{e^{- t}}{1 + e^{t}} & - \frac{1}{1 + e^{t}} \cdot - 1 \\ - \frac{1}{1 + e^{t}} (- e^{t}) & - \frac{1}{1 + e^{t}} (- e^{t}) \end{array}]$

Step 10.2

Multiply $- \frac{1}{1 + e^{t}} \cdot - 1$ .

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Step 10.2.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{e^{- t}}{1 + e^{t}} & 1 \frac{1}{1 + e^{t}} \\ - \frac{1}{1 + e^{t}} (- e^{t}) & - \frac{1}{1 + e^{t}} (- e^{t}) \end{array}]$

Step 10.2.2

Multiply $\frac{1}{1 + e^{t}}$ by $1$ .

$[\begin{array}{c} - \frac{e^{- t}}{1 + e^{t}} & \frac{1}{1 + e^{t}} \\ - \frac{1}{1 + e^{t}} (- e^{t}) & - \frac{1}{1 + e^{t}} (- e^{t}) \end{array}]$

Step 10.3

Multiply $- \frac{1}{1 + e^{t}} (- e^{t})$ .

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Step 10.3.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{e^{- t}}{1 + e^{t}} & \frac{1}{1 + e^{t}} \\ 1 \frac{1}{1 + e^{t}} e^{t} & - \frac{1}{1 + e^{t}} (- e^{t}) \end{array}]$

Step 10.3.2

Multiply $\frac{1}{1 + e^{t}}$ by $1$ .

$[\begin{array}{c} - \frac{e^{- t}}{1 + e^{t}} & \frac{1}{1 + e^{t}} \\ \frac{1}{1 + e^{t}} e^{t} & - \frac{1}{1 + e^{t}} (- e^{t}) \end{array}]$

Step 10.3.3

Combine $\frac{1}{1 + e^{t}}$ and $e^{t}$ .

$[\begin{array}{c} - \frac{e^{- t}}{1 + e^{t}} & \frac{1}{1 + e^{t}} \\ \frac{e^{t}}{1 + e^{t}} & - \frac{1}{1 + e^{t}} (- e^{t}) \end{array}]$

Step 10.4

Multiply $- \frac{1}{1 + e^{t}} (- e^{t})$ .

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Step 10.4.1

Multiply $- 1$ by $- 1$ .

$[\begin{array}{c} - \frac{e^{- t}}{1 + e^{t}} & \frac{1}{1 + e^{t}} \\ \frac{e^{t}}{1 + e^{t}} & 1 \frac{1}{1 + e^{t}} e^{t} \end{array}]$

Step 10.4.2

Multiply $\frac{1}{1 + e^{t}}$ by $1$ .

$[\begin{array}{c} - \frac{e^{- t}}{1 + e^{t}} & \frac{1}{1 + e^{t}} \\ \frac{e^{t}}{1 + e^{t}} & \frac{1}{1 + e^{t}} e^{t} \end{array}]$

Step 10.4.3

Combine $\frac{1}{1 + e^{t}}$ and $e^{t}$ .

$[\begin{array}{c} - \frac{e^{- t}}{1 + e^{t}} & \frac{1}{1 + e^{t}} \\ \frac{e^{t}}{1 + e^{t}} & \frac{e^{t}}{1 + e^{t}} \end{array}]$