Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 3 & 3 & 2 0 & 4 & 2 3 & 2 & 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 3 & 3 & 2 \\ 0 & 4 & 2 \\ 3 & 2 & 1 \end{array}]$

Step 1

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 column

1

$1$ by its cofactor and add.

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Step 1.1

Consider the corresponding sign chart.

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

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

Step 1.2

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

-

$-$ position on the sign chart.

Step 1.3

The minor for

a_{11}

$a_{11}$ is the determinant with row

1

$1$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 4 & 2 2 & 1 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 4 & 2 \\ 2 & 1 \end{array} |$

Step 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

3 ∣ ∣ ∣ \begin{matrix} 4 & 2 2 & 1 \end{matrix} ∣ ∣ ∣

$3 | \begin{array}{c} 4 & 2 \\ 2 & 1 \end{array} |$

Step 1.5

The minor for

a_{21}

$a_{21}$ is the determinant with row

2

$2$ and column

1

$1$ deleted.

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

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

Step 1.6

Multiply element

a_{21}

$a_{21}$ by its cofactor.

0 ∣ ∣ ∣ \begin{matrix} 3 & 2 2 & 1 \end{matrix} ∣ ∣ ∣

$0 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} |$

Step 1.7

The minor for

a_{31}

$a_{31}$ is the determinant with row

3

$3$ and column

1

$1$ deleted.

∣ ∣ ∣ \begin{matrix} 3 & 2 4 & 2 \end{matrix} ∣ ∣ ∣

$| \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$

Step 1.8

Multiply element

a_{31}

$a_{31}$ by its cofactor.

3 ∣ ∣ ∣ \begin{matrix} 3 & 2 4 & 2 \end{matrix} ∣ ∣ ∣

$3 | \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$

Step 1.9

Add the terms together.

3 ∣ ∣ ∣ \begin{matrix} 4 & 2 2 & 1 \end{matrix} ∣ ∣ ∣ + 0 ∣ ∣ ∣ \begin{matrix} 3 & 2 2 & 1 \end{matrix} ∣ ∣ ∣ + 3 ∣ ∣ ∣ \begin{matrix} 3 & 2 4 & 2 \end{matrix} ∣ ∣ ∣

$3 | \begin{array}{c} 4 & 2 \\ 2 & 1 \end{array} | + 0 | \begin{array}{c} 3 & 2 \\ 2 & 1 \end{array} | + 3 | \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$

Step 2

Multiply

$0$ by

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

$3 | \begin{array}{c} 4 & 2 \\ 2 & 1 \end{array} | + 0 + 3 | \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$

Step 3

Evaluate

$| \begin{array}{c} 4 & 2 \\ 2 & 1 \end{array} |$ .

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Step 3.1

The determinant of a

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

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

$3 (4 \cdot 1 - 2 \cdot 2) + 0 + 3 | \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$

Step 3.2

Simplify the determinant.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply

$4$ by

$1$ .

$3 (4 - 2 \cdot 2) + 0 + 3 | \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$

Step 3.2.1.2

Multiply

$- 2$ by

$2$ .

$3 (4 - 4) + 0 + 3 | \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$

Step 3.2.2

Subtract

$4$ from

$4$ .

$3 \cdot 0 + 0 + 3 | \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$

Step 4

Evaluate

$| \begin{array}{c} 3 & 2 \\ 4 & 2 \end{array} |$ .

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Step 4.1

The determinant of a

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

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

$3 \cdot 0 + 0 + 3 (3 \cdot 2 - 4 \cdot 2)$

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

$3$ by

$2$ .

$3 \cdot 0 + 0 + 3 (6 - 4 \cdot 2)$

Step 4.2.1.2

Multiply

$- 4$ by

$2$ .

$3 \cdot 0 + 0 + 3 (6 - 8)$

Step 4.2.2

Subtract

$8$ from

$6$ .

$3 \cdot 0 + 0 + 3 \cdot - 2$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply

$3$ by

$0$ .

$0 + 0 + 3 \cdot - 2$

Step 5.1.2

Multiply

$3$ by

$- 2$ .

$0 + 0 - 6$

Step 5.2

Add

$0$ and

$0$ .

$0 - 6$

Step 5.3

Subtract

$6$ from

$0$ .

$- 6$

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