Linear Algebra Examples

$[\begin{array}{c} 3 e^{t} & e^{2 t} \\ 2 e^{t} & 2 e^{2 t} \end{array}]$

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

$3 e^{t} (2 e^{2 t}) - 2 e^{t} e^{2 t}$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 2 e^{t} e^{2 t} - 2 e^{t} e^{2 t}$

Step 2.2.1.2

Multiply $e^{t}$ by $e^{2 t}$ by adding the exponents.

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Step 2.2.1.2.1

Move $e^{2 t}$ .

$3 \cdot 2 (e^{2 t} e^{t}) - 2 e^{t} e^{2 t}$

Step 2.2.1.2.2

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

$3 \cdot 2 e^{2 t + t} - 2 e^{t} e^{2 t}$

Step 2.2.1.2.3

Add $2 t$ and $t$ .

$3 \cdot 2 e^{3 t} - 2 e^{t} e^{2 t}$

Step 2.2.1.3

Multiply $3$ by $2$ .

$6 e^{3 t} - 2 e^{t} e^{2 t}$

Step 2.2.1.4

Multiply $e^{t}$ by $e^{2 t}$ by adding the exponents.

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Step 2.2.1.4.1

Move $e^{2 t}$ .

$6 e^{3 t} - 2 (e^{2 t} e^{t})$

Step 2.2.1.4.2

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

$6 e^{3 t} - 2 e^{2 t + t}$

Step 2.2.1.4.3

Add $2 t$ and $t$ .

$6 e^{3 t} - 2 e^{3 t}$

Step 2.2.2

Subtract $2 e^{3 t}$ from $6 e^{3 t}$ .

$4 e^{3 t}$

Step 3

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

Step 4

Substitute the known values into the formula for the inverse.

$\frac{1}{4 e^{3 t}} [\begin{array}{c} 2 e^{2 t} & - e^{2 t} \\ - 2 e^{t} & 3 e^{t} \end{array}]$

Step 5

Multiply $\frac{1}{4 e^{3 t}}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{4 e^{3 t}} (2 e^{2 t}) & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6

Simplify each element in the matrix.

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Step 6.1

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} 2 \frac{1}{4 e^{3 t}} e^{2 t} & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.2

Cancel the common factor of $2$ .

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Step 6.2.1

Factor $2$ out of $4 e^{3 t}$ .

$[\begin{array}{c} 2 \frac{1}{2 (2 e^{3 t})} e^{2 t} & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.2.2

Cancel the common factor.

$[\begin{array}{c} 2 \frac{1}{2 (2 e^{3 t})} e^{2 t} & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{1}{2 e^{3 t}} e^{2 t} & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.3

Combine $\frac{1}{2 e^{3 t}}$ and $e^{2 t}$ .

$[\begin{array}{c} \frac{e^{2 t}}{2 e^{3 t}} & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.4

Cancel the common factor of $e^{2 t}$ and $e^{3 t}$ .

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Step 6.4.1

Factor $e^{3 t}$ out of $e^{2 t}$ .

$[\begin{array}{c} \frac{e^{3 t} e^{- t}}{2 e^{3 t}} & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $e^{3 t}$ out of $2 e^{3 t}$ .

$[\begin{array}{c} \frac{e^{3 t} e^{- t}}{e^{3 t} \cdot 2} & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.4.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{e^{3 t} e^{- t}}{e^{3 t} \cdot 2} & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.4.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{e^{- t}}{2} & \frac{1}{4 e^{3 t}} (- e^{2 t}) \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.5

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{1}{4 e^{3 t}} e^{2 t} \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.6

Combine $e^{2 t}$ and $\frac{1}{4 e^{3 t}}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{2 t}}{4 e^{3 t}} \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.7

Cancel the common factor of $e^{2 t}$ and $e^{3 t}$ .

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Step 6.7.1

Factor $e^{3 t}$ out of $e^{2 t}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{3 t} e^{- t}}{4 e^{3 t}} \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.7.2

Cancel the common factors.

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Step 6.7.2.1

Factor $e^{3 t}$ out of $4 e^{3 t}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{3 t} e^{- t}}{e^{3 t} \cdot 4} \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.7.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{3 t} e^{- t}}{e^{3 t} \cdot 4} \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.7.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ \frac{1}{4 e^{3 t}} (- 2 e^{t}) & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.8

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - 2 \frac{1}{4 e^{3 t}} e^{t} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.9

Cancel the common factor of $2$ .

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Step 6.9.1

Factor $2$ out of $- 2$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ 2 (- 1) \frac{1}{4 e^{3 t}} e^{t} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.9.2

Factor $2$ out of $4 e^{3 t}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ 2 (- 1) \frac{1}{2 (2 e^{3 t})} e^{t} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.9.3

Cancel the common factor.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ 2 \cdot - 1 \frac{1}{2 (2 e^{3 t})} e^{t} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.9.4

Rewrite the expression.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{1}{2 e^{3 t}} e^{t} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.10

Combine $e^{t}$ and $\frac{1}{2 e^{3 t}}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{t}}{2 e^{3 t}} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.11

Cancel the common factor of $e^{t}$ and $e^{3 t}$ .

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Step 6.11.1

Factor $e^{3 t}$ out of $e^{t}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{3 t} e^{- 2 t}}{2 e^{3 t}} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.11.2

Cancel the common factors.

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Step 6.11.2.1

Factor $e^{3 t}$ out of $2 e^{3 t}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{3 t} e^{- 2 t}}{e^{3 t} \cdot 2} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.11.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{3 t} e^{- 2 t}}{e^{3 t} \cdot 2} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.11.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{- 2 t}}{2} & \frac{1}{4 e^{3 t}} (3 e^{t}) \end{array}]$

Step 6.12

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{- 2 t}}{2} & 3 \frac{1}{4 e^{3 t}} e^{t} \end{array}]$

Step 6.13

Combine $3$ and $\frac{1}{4 e^{3 t}}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{- 2 t}}{2} & \frac{3}{4 e^{3 t}} e^{t} \end{array}]$

Step 6.14

Combine $\frac{3}{4 e^{3 t}}$ and $e^{t}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{- 2 t}}{2} & \frac{3 e^{t}}{4 e^{3 t}} \end{array}]$

Step 6.15

Cancel the common factor of $e^{t}$ and $e^{3 t}$ .

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Step 6.15.1

Factor $e^{3 t}$ out of $3 e^{t}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{- 2 t}}{2} & \frac{e^{3 t} (3 e^{- 2 t})}{4 e^{3 t}} \end{array}]$

Step 6.15.2

Cancel the common factors.

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Step 6.15.2.1

Factor $e^{3 t}$ out of $4 e^{3 t}$ .

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{- 2 t}}{2} & \frac{e^{3 t} (3 e^{- 2 t})}{e^{3 t} \cdot 4} \end{array}]$

Step 6.15.2.2

Cancel the common factor.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{- 2 t}}{2} & \frac{e^{3 t} (3 e^{- 2 t})}{e^{3 t} \cdot 4} \end{array}]$

Step 6.15.2.3

Rewrite the expression.

$[\begin{array}{c} \frac{e^{- t}}{2} & - \frac{e^{- t}}{4} \\ - \frac{e^{- 2 t}}{2} & \frac{3 e^{- 2 t}}{4} \end{array}]$