Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 2 - 1 1 \end{matrix} ⎤ ⎥ ⎦ \times ⎡ ⎢ ⎣ \begin{matrix} 5 - 3 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 2 \\ - 1 \\ 1 \end{array}] \times [\begin{array}{c} 5 \\ - 3 \\ 1 \end{array}]$

Step 1

The cross product of two vectors

a⃗

$a⃗$ and

b⃗

$b⃗$ can be written as a determinant with the standard unit vectors from

R^{3}

$ℝ^{3}$ and the elements of the given vectors.

a⃗ \times b⃗ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} î & ĵ & k̂ a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} \end{matrix} ∣ ∣ ∣ ∣ ∣

$a⃗ \times b⃗ = | \begin{array}{c} î & ĵ & k̂ \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array} |$

Step 2

Set up the determinant with the given values.

∣ ∣ ∣ ∣ ∣ \begin{matrix} î & ĵ & k̂ 2 & - 1 & 1 5 & - 3 & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

$| \begin{array}{c} î & ĵ & k̂ \\ 2 & - 1 & 1 \\ 5 & - 3 & 1 \end{array} |$

Step 3

Choose the row or column with the most

0

$0$ elements. If there are no

0

$0$ elements choose any row or column. Multiply every element in row

1

$1$ by its cofactor and add.

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Step 3.1

Consider the corresponding sign chart.

∣ ∣ ∣ ∣ \begin{matrix} + & - & + - & + & - + & - & + \end{matrix} ∣ ∣ ∣ ∣

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 3.2

The cofactor is the minor with the sign changed if the indices match a

-

$-$ position on the sign chart.

Step 3.3

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} - 1 & 1 - 3 & 1 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} - 1 & 1 \\ - 3 & 1 \end{array} |$

Step 3.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

∣ ∣ ∣ \begin{matrix} - 1 & 1 - 3 & 1 \end{matrix} ∣ ∣ ∣ î

$| \begin{array}{c} - 1 & 1 \\ - 3 & 1 \end{array} | î$

Step 3.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} |$

Step 3.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

- ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ

$- | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ$

Step 3.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} |$

Step 3.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$| \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 3.9

Add the terms together.

∣ ∣ ∣ \begin{matrix} - 1 & 1 - 3 & 1 \end{matrix} ∣ ∣ ∣ î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$| \begin{array}{c} - 1 & 1 \\ - 3 & 1 \end{array} | î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

∣ ∣ ∣ \begin{matrix} - 1 & 1 - 3 & 1 \end{matrix} ∣ ∣ ∣ î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$| \begin{array}{c} - 1 & 1 \\ - 3 & 1 \end{array} | î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 4

Evaluate

∣ ∣ ∣ \begin{matrix} - 1 & 1 - 3 & 1 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} - 1 & 1 \\ - 3 & 1 \end{array} |$ .

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Step 4.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

(- 1 \cdot 1 - (- 3 \cdot 1)) î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$(- 1 \cdot 1 - (- 3 \cdot 1)) î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply

- 1

$- 1$ by

1

$1$ .

(- 1 - (- 3 \cdot 1)) î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$(- 1 - (- 3 \cdot 1)) î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 4.2.1.2

Multiply

- (- 3 \cdot 1)

$- (- 3 \cdot 1)$ .

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Step 4.2.1.2.1

Multiply

- 3

$- 3$ by

1

$1$ .

(- 1 - - 3) î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$(- 1 - - 3) î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 4.2.1.2.2

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

(- 1 + 3) î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$(- 1 + 3) î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

(- 1 + 3) î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$(- 1 + 3) î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

(- 1 + 3) î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$(- 1 + 3) î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 4.2.2

Add

- 1

$- 1$ and

3

$3$ .

2 î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$2 î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

2 î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$2 î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

2 î - ∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣ ĵ + ∣ ∣ ∣ \begin{matrix} 2 & - 1 5 & - 3 \end{matrix} ∣ ∣ ∣ k̂

$2 î - | \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} | ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 5

Evaluate

∣ ∣ ∣ \begin{matrix} 2 & 1 5 & 1 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 2 & 1 \\ 5 & 1 \end{array} |$ .

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Step 5.1

The determinant of a

2 \times 2

$2 \times 2$ matrix can be found using the formula

∣ ∣ ∣ \begin{matrix} a & b c & d \end{matrix} ∣ ∣ ∣ = a d - c b

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

$2 î - (2 \cdot 1 - 5 \cdot 1) ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 5.2

Simplify the determinant.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply

$2$ by

$1$ .

$2 î - (2 - 5 \cdot 1) ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 5.2.1.2

Multiply

$- 5$ by

$1$ .

$2 î - (2 - 5) ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 5.2.2

Subtract

$5$ from

$2$ .

$2 î - - 3 ĵ + | \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} | k̂$

Step 6

Evaluate

$| \begin{array}{c} 2 & - 1 \\ 5 & - 3 \end{array} |$ .

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Step 6.1

The determinant of a

$2 \times 2$ matrix can be found using the formula

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

$2 î - - 3 ĵ + (2 \cdot - 3 - 5 \cdot - 1) k̂$

Step 6.2

Simplify the determinant.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply

$2$ by

$- 3$ .

$2 î - - 3 ĵ + (- 6 - 5 \cdot - 1) k̂$

Step 6.2.1.2

Multiply

$- 5$ by

$- 1$ .

$2 î - - 3 ĵ + (- 6 + 5) k̂$

Step 6.2.2

Add

$- 6$ and

$5$ .

$2 î - - 3 ĵ - k̂$

Step 7

Multiply

$- 1$ by

$- 3$ .

$2 î + 3 ĵ - k̂$

Step 8

Rewrite the answer.

$[\begin{array}{c} 2 \\ 3 \\ - 1 \end{array}]$

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