Linear Algebra Examples

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

Step 1

To write $s$ as a fraction with a common denominator, multiply by $\frac{s + 1}{s + 1}$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Apply the distributive property.

$[\begin{array}{c} 2 s + 3 & - 3 \\ \frac{4 + s \cdot s + s \cdot 1}{s + 1} + 2 & s^{2} \end{array}]$

Step 3.2

Multiply $s$ by $s$ .

$[\begin{array}{c} 2 s + 3 & - 3 \\ \frac{4 + s^{2} + s \cdot 1}{s + 1} + 2 & s^{2} \end{array}]$

Step 3.3

Multiply $s$ by $1$ .

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

Step 3.4

Reorder terms.

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

Step 4

To write $2$ as a fraction with a common denominator, multiply by $\frac{s + 1}{s + 1}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$[\begin{array}{c} 2 s + 3 & - 3 \\ \frac{s^{2} + s + 4 + 2 s + 2 \cdot 1}{s + 1} & s^{2} \end{array}]$

Step 6.2

Multiply $2$ by $1$ .

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

Step 6.3

Add $s$ and $2 s$ .

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

Step 6.4

Add $4$ and $2$ .

$[\begin{array}{c} 2 s + 3 & - 3 \\ \frac{s^{2} + 3 s + 6}{s + 1} & s^{2} \end{array}]$

Step 7

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

$(2 s + 3) s^{2} - \frac{s^{2} + 3 s + 6}{s + 1} \cdot - 3$

Step 8

Simplify the determinant.

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Step 8.1

Simplify each term.

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Step 8.1.1

Apply the distributive property.

$2 s \cdot s^{2} + 3 s^{2} - \frac{s^{2} + 3 s + 6}{s + 1} \cdot - 3$

Step 8.1.2

Multiply $s$ by $s^{2}$ by adding the exponents.

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Step 8.1.2.1

Move $s^{2}$ .

$2 (s^{2} s) + 3 s^{2} - \frac{s^{2} + 3 s + 6}{s + 1} \cdot - 3$

Step 8.1.2.2

Multiply $s^{2}$ by $s$ .

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Step 8.1.2.2.1

Raise $s$ to the power of $1$ .

$2 (s^{2} s^{1}) + 3 s^{2} - \frac{s^{2} + 3 s + 6}{s + 1} \cdot - 3$

Step 8.1.2.2.2

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

$2 s^{2 + 1} + 3 s^{2} - \frac{s^{2} + 3 s + 6}{s + 1} \cdot - 3$

Step 8.1.2.3

Add $2$ and $1$ .

$2 s^{3} + 3 s^{2} - \frac{s^{2} + 3 s + 6}{s + 1} \cdot - 3$

Step 8.1.3

Multiply $- \frac{s^{2} + 3 s + 6}{s + 1} \cdot - 3$ .

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Step 8.1.3.1

Multiply $- 3$ by $- 1$ .

$2 s^{3} + 3 s^{2} + 3 \frac{s^{2} + 3 s + 6}{s + 1}$

Step 8.1.3.2

Combine $3$ and $\frac{s^{2} + 3 s + 6}{s + 1}$ .

$2 s^{3} + 3 s^{2} + \frac{3 (s^{2} + 3 s + 6)}{s + 1}$

Step 8.2

To write $2 s^{3}$ as a fraction with a common denominator, multiply by $\frac{s + 1}{s + 1}$ .

$3 s^{2} + \frac{2 s^{3} (s + 1)}{s + 1} + \frac{3 (s^{2} + 3 s + 6)}{s + 1}$

Step 8.3

Combine the numerators over the common denominator.

$3 s^{2} + \frac{2 s^{3} (s + 1) + 3 (s^{2} + 3 s + 6)}{s + 1}$

Step 8.4

Simplify the numerator.

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Step 8.4.1

Apply the distributive property.

$3 s^{2} + \frac{2 s^{3} s + 2 s^{3} \cdot 1 + 3 (s^{2} + 3 s + 6)}{s + 1}$

Step 8.4.2

Multiply $s^{3}$ by $s$ by adding the exponents.

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Step 8.4.2.1

Move $s$ .

$3 s^{2} + \frac{2 (s \cdot s^{3}) + 2 s^{3} \cdot 1 + 3 (s^{2} + 3 s + 6)}{s + 1}$

Step 8.4.2.2

Multiply $s$ by $s^{3}$ .

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Step 8.4.2.2.1

Raise $s$ to the power of $1$ .

$3 s^{2} + \frac{2 (s^{1} s^{3}) + 2 s^{3} \cdot 1 + 3 (s^{2} + 3 s + 6)}{s + 1}$

Step 8.4.2.2.2

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

$3 s^{2} + \frac{2 s^{1 + 3} + 2 s^{3} \cdot 1 + 3 (s^{2} + 3 s + 6)}{s + 1}$

Step 8.4.2.3

Add $1$ and $3$ .

$3 s^{2} + \frac{2 s^{4} + 2 s^{3} \cdot 1 + 3 (s^{2} + 3 s + 6)}{s + 1}$

Step 8.4.3

Multiply $2$ by $1$ .

$3 s^{2} + \frac{2 s^{4} + 2 s^{3} + 3 (s^{2} + 3 s + 6)}{s + 1}$

Step 8.4.4

Apply the distributive property.

$3 s^{2} + \frac{2 s^{4} + 2 s^{3} + 3 s^{2} + 3 (3 s) + 3 \cdot 6}{s + 1}$

Step 8.4.5

Simplify.

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Step 8.4.5.1

Multiply $3$ by $3$ .

$3 s^{2} + \frac{2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 3 \cdot 6}{s + 1}$

Step 8.4.5.2

Multiply $3$ by $6$ .

$3 s^{2} + \frac{2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.5

To write $3 s^{2}$ as a fraction with a common denominator, multiply by $\frac{s + 1}{s + 1}$ .

$\frac{3 s^{2} (s + 1)}{s + 1} + \frac{2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.6

Combine the numerators over the common denominator.

$\frac{3 s^{2} (s + 1) + 2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.7

Simplify the numerator.

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Step 8.7.1

Apply the distributive property.

$\frac{3 s^{2} s + 3 s^{2} \cdot 1 + 2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.7.2

Multiply $s^{2}$ by $s$ by adding the exponents.

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Step 8.7.2.1

Move $s$ .

$\frac{3 (s \cdot s^{2}) + 3 s^{2} \cdot 1 + 2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.7.2.2

Multiply $s$ by $s^{2}$ .

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Step 8.7.2.2.1

Raise $s$ to the power of $1$ .

$\frac{3 (s^{1} s^{2}) + 3 s^{2} \cdot 1 + 2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.7.2.2.2

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

$\frac{3 s^{1 + 2} + 3 s^{2} \cdot 1 + 2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.7.2.3

Add $1$ and $2$ .

$\frac{3 s^{3} + 3 s^{2} \cdot 1 + 2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.7.3

Multiply $3$ by $1$ .

$\frac{3 s^{3} + 3 s^{2} + 2 s^{4} + 2 s^{3} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.7.4

Add $3 s^{3}$ and $2 s^{3}$ .

$\frac{5 s^{3} + 3 s^{2} + 2 s^{4} + 3 s^{2} + 9 s + 18}{s + 1}$

Step 8.7.5

Add $3 s^{2}$ and $3 s^{2}$ .

$\frac{5 s^{3} + 2 s^{4} + 6 s^{2} + 9 s + 18}{s + 1}$

Step 8.7.6

Reorder terms.

$\frac{2 s^{4} + 5 s^{3} + 6 s^{2} + 9 s + 18}{s + 1}$