Linear Algebra Examples

\frac{- 4}{3 + i}

$\frac{- 4}{3 + i}$

Step 1

Multiply the numerator and denominator of

\frac{- 4}{3 + 1 i}

$\frac{- 4}{3 + 1 i}$ by the conjugate of

3 + 1 i

$3 + 1 i$ to make the denominator real.

\frac{- 4}{3 + 1 i} \cdot \frac{3 - i}{3 - i}

$\frac{- 4}{3 + 1 i} \cdot \frac{3 - i}{3 - i}$

Step 2

Multiply.

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Step 2.1

Combine.

\frac{- 4 (3 - i)}{(3 + 1 i) (3 - i)}

$\frac{- 4 (3 - i)}{(3 + 1 i) (3 - i)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

\frac{- 4 \cdot 3 - 4 (- i)}{(3 + 1 i) (3 - i)}

$\frac{- 4 \cdot 3 - 4 (- i)}{(3 + 1 i) (3 - i)}$

Step 2.2.2

Multiply

- 4

$- 4$ by

3

$3$ .

\frac{- 12 - 4 (- i)}{(3 + 1 i) (3 - i)}

$\frac{- 12 - 4 (- i)}{(3 + 1 i) (3 - i)}$

Step 2.2.3

Multiply

- 1

$- 1$ by

- 4

$- 4$ .

\frac{- 12 + 4 i}{(3 + 1 i) (3 - i)}

$\frac{- 12 + 4 i}{(3 + 1 i) (3 - i)}$

\frac{- 12 + 4 i}{(3 + 1 i) (3 - i)}

$\frac{- 12 + 4 i}{(3 + 1 i) (3 - i)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand

(3 + 1 i) (3 - i)

$(3 + 1 i) (3 - i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

\frac{- 12 + 4 i}{3 (3 - i) + 1 i (3 - i)}

$\frac{- 12 + 4 i}{3 (3 - i) + 1 i (3 - i)}$

Step 2.3.1.2

Apply the distributive property.

\frac{- 12 + 4 i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}

$\frac{- 12 + 4 i}{3 \cdot 3 + 3 (- i) + 1 i (3 - i)}$

Step 2.3.1.3

Apply the distributive property.

\frac{- 12 + 4 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}

$\frac{- 12 + 4 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

\frac{- 12 + 4 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}

$\frac{- 12 + 4 i}{3 \cdot 3 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply

3

$3$ by

3

$3$ .

\frac{- 12 + 4 i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}

$\frac{- 12 + 4 i}{9 + 3 (- i) + 1 i \cdot 3 + 1 i (- i)}$

Step 2.3.2.2

Multiply

- 1

$- 1$ by

3

$3$ .

\frac{- 12 + 4 i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}

$\frac{- 12 + 4 i}{9 - 3 i + 1 i \cdot 3 + 1 i (- i)}$

Step 2.3.2.3

Multiply

3

$3$ by

1

$1$ .

\frac{- 12 + 4 i}{9 - 3 i + 3 i + 1 i (- i)}

$\frac{- 12 + 4 i}{9 - 3 i + 3 i + 1 i (- i)}$

Step 2.3.2.4

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{- 12 + 4 i}{9 - 3 i + 3 i - i i}

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - i i}$

Step 2.3.2.5

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 12 + 4 i}{9 - 3 i + 3 i - (i^{1} i)}

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - (i^{1} i)}$

Step 2.3.2.6

Raise

i

$i$ to the power of

1

$1$ .

\frac{- 12 + 4 i}{9 - 3 i + 3 i - (i^{1} i^{1})}

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - (i^{1} i^{1})}$

Step 2.3.2.7

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{- 12 + 4 i}{9 - 3 i + 3 i - i^{1 + 1}}

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - i^{1 + 1}}$

Step 2.3.2.8

Add

1

$1$ and

1

$1$ .

\frac{- 12 + 4 i}{9 - 3 i + 3 i - i^{2}}

$\frac{- 12 + 4 i}{9 - 3 i + 3 i - i^{2}}$

Step 2.3.2.9

Add

- 3 i

$- 3 i$ and

3 i

$3 i$ .

\frac{- 12 + 4 i}{9 + 0 - i^{2}}

$\frac{- 12 + 4 i}{9 + 0 - i^{2}}$

Step 2.3.2.10

Add

9

$9$ and

0

$0$ .

\frac{- 12 + 4 i}{9 - i^{2}}

$\frac{- 12 + 4 i}{9 - i^{2}}$

\frac{- 12 + 4 i}{9 - i^{2}}

$\frac{- 12 + 4 i}{9 - i^{2}}$

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

\frac{- 12 + 4 i}{9 - - 1}

$\frac{- 12 + 4 i}{9 - - 1}$

Step 2.3.3.2

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{- 12 + 4 i}{9 + 1}

$\frac{- 12 + 4 i}{9 + 1}$

\frac{- 12 + 4 i}{9 + 1}

$\frac{- 12 + 4 i}{9 + 1}$

Step 2.3.4

Add

9

$9$ and

1

$1$ .

\frac{- 12 + 4 i}{10}

$\frac{- 12 + 4 i}{10}$

\frac{- 12 + 4 i}{10}

$\frac{- 12 + 4 i}{10}$

\frac{- 12 + 4 i}{10}

$\frac{- 12 + 4 i}{10}$

Step 3

Cancel the common factor of

- 12 + 4 i

$- 12 + 4 i$ and

10

$10$ .

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Step 3.1

Factor

2

$2$ out of

- 12

$- 12$ .

\frac{2 (- 6) + 4 i}{10}

$\frac{2 (- 6) + 4 i}{10}$

Step 3.2

Factor

2

$2$ out of

4 i

$4 i$ .

\frac{2 (- 6) + 2 (2 i)}{10}

$\frac{2 (- 6) + 2 (2 i)}{10}$

Step 3.3

Factor

2

$2$ out of

2 (- 6) + 2 (2 i)

$2 (- 6) + 2 (2 i)$ .

\frac{2 (- 6 + 2 i)}{10}

$\frac{2 (- 6 + 2 i)}{10}$

Step 3.4

Cancel the common factors.

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Step 3.4.1

Factor

2

$2$ out of

10

$10$ .

\frac{2 (- 6 + 2 i)}{2 \cdot 5}

$\frac{2 (- 6 + 2 i)}{2 \cdot 5}$

Step 3.4.2

Cancel the common factor.

\frac{2 (- 6 + 2 i)}{2 \cdot 5}

$\frac{2 (- 6 + 2 i)}{2 \cdot 5}$

Step 3.4.3

Rewrite the expression.

\frac{- 6 + 2 i}{5}

$\frac{- 6 + 2 i}{5}$

\frac{- 6 + 2 i}{5}

$\frac{- 6 + 2 i}{5}$

\frac{- 6 + 2 i}{5}

$\frac{- 6 + 2 i}{5}$

Step 4

Split the fraction

\frac{- 6 + 2 i}{5}

$\frac{- 6 + 2 i}{5}$ into two fractions.

\frac{- 6}{5} + \frac{2 i}{5}

$\frac{- 6}{5} + \frac{2 i}{5}$

Step 5

Move the negative in front of the fraction.

- \frac{6}{5} + \frac{2 i}{5}

$- \frac{6}{5} + \frac{2 i}{5}$

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