Linear Algebra Examples

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

Step 1

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

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

Step 2

Simplify the determinant.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$2 e^{3 x} e^{2 x} - 3 e^{3 x} e^{2 x}$

Step 2.1.2

Multiply $e^{3 x}$ by $e^{2 x}$ by adding the exponents.

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Step 2.1.2.1

Move $e^{2 x}$ .

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

Step 2.1.2.2

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

$2 e^{2 x + 3 x} - 3 e^{3 x} e^{2 x}$

Step 2.1.2.3

Add $2 x$ and $3 x$ .

$2 e^{5 x} - 3 e^{3 x} e^{2 x}$

Step 2.1.3

Multiply $e^{3 x}$ by $e^{2 x}$ by adding the exponents.

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Step 2.1.3.1

Move $e^{2 x}$ .

$2 e^{5 x} - 3 (e^{2 x} e^{3 x})$

Step 2.1.3.2

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

$2 e^{5 x} - 3 e^{2 x + 3 x}$

Step 2.1.3.3

Add $2 x$ and $3 x$ .

$2 e^{5 x} - 3 e^{5 x}$

Step 2.2

Subtract $3 e^{5 x}$ from $2 e^{5 x}$ .

$- e^{5 x}$