Linear Algebra Examples

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

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

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 row

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} - a & 1 - 2 & 3 \end{matrix} ∣ ∣ ∣

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

Step 1.4

Multiply element

a_{11}

$a_{11}$ by its cofactor.

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

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

Step 1.5

The minor for

a_{12}

$a_{12}$ is the determinant with row

1

$1$ and column

2

$2$ deleted.

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

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

Step 1.6

Multiply element

a_{12}

$a_{12}$ by its cofactor.

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

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

Step 1.7

The minor for

a_{13}

$a_{13}$ is the determinant with row

1

$1$ and column

3

$3$ deleted.

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

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

Step 1.8

Multiply element

a_{13}

$a_{13}$ by its cofactor.

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

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

Step 1.9

Add the terms together.

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

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

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

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

Step 2

Multiply

0

$0$ by

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

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

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

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

Step 3

Evaluate

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

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

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Step 3.1

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

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

0 + 1 (3 \cdot 3 - 1 \cdot 1) + a ∣ ∣ ∣ \begin{matrix} 3 & - a 1 & - 2 \end{matrix} ∣ ∣ ∣

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

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

3

$3$ by

3

$3$ .

0 + 1 (9 - 1 \cdot 1) + a ∣ ∣ ∣ \begin{matrix} 3 & - a 1 & - 2 \end{matrix} ∣ ∣ ∣

$0 + 1 (9 - 1 \cdot 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Step 3.2.1.2

Multiply

- 1

$- 1$ by

1

$1$ .

0 + 1 (9 - 1) + a ∣ ∣ ∣ \begin{matrix} 3 & - a 1 & - 2 \end{matrix} ∣ ∣ ∣

$0 + 1 (9 - 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

0 + 1 (9 - 1) + a ∣ ∣ ∣ \begin{matrix} 3 & - a 1 & - 2 \end{matrix} ∣ ∣ ∣

$0 + 1 (9 - 1) + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Step 3.2.2

Subtract

1

$1$ from

9

$9$ .

0 + 1 \cdot 8 + a ∣ ∣ ∣ \begin{matrix} 3 & - a 1 & - 2 \end{matrix} ∣ ∣ ∣

$0 + 1 \cdot 8 + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

0 + 1 \cdot 8 + a ∣ ∣ ∣ \begin{matrix} 3 & - a 1 & - 2 \end{matrix} ∣ ∣ ∣

$0 + 1 \cdot 8 + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

0 + 1 \cdot 8 + a ∣ ∣ ∣ \begin{matrix} 3 & - a 1 & - 2 \end{matrix} ∣ ∣ ∣

$0 + 1 \cdot 8 + a | \begin{array}{c} 3 & - a \\ 1 & - 2 \end{array} |$

Step 4

Evaluate

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

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

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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

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

0 + 1 \cdot 8 + a (3 \cdot - 2 - - a)

$0 + 1 \cdot 8 + a (3 \cdot - 2 - - a)$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply

3

$3$ by

- 2

$- 2$ .

0 + 1 \cdot 8 + a (- 6 - - a)

$0 + 1 \cdot 8 + a (- 6 - - a)$

Step 4.2.2

Multiply

- - a

$- - a$ .

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Step 4.2.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

0 + 1 \cdot 8 + a (- 6 + 1 a)

$0 + 1 \cdot 8 + a (- 6 + 1 a)$

Step 4.2.2.2

Multiply

a

$a$ by

1

$1$ .

0 + 1 \cdot 8 + a (- 6 + a)

$0 + 1 \cdot 8 + a (- 6 + a)$

0 + 1 \cdot 8 + a (- 6 + a)

$0 + 1 \cdot 8 + a (- 6 + a)$

0 + 1 \cdot 8 + a (- 6 + a)

$0 + 1 \cdot 8 + a (- 6 + a)$

0 + 1 \cdot 8 + a (- 6 + a)

$0 + 1 \cdot 8 + a (- 6 + a)$

Step 5

Simplify the determinant.

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Step 5.1

Add

0

$0$ and

1 \cdot 8

$1 \cdot 8$ .

1 \cdot 8 + a (- 6 + a)

$1 \cdot 8 + a (- 6 + a)$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply

8

$8$ by

1

$1$ .

8 + a (- 6 + a)

$8 + a (- 6 + a)$

Step 5.2.2

Apply the distributive property.

8 + a \cdot - 6 + a \cdot a

$8 + a \cdot - 6 + a \cdot a$

Step 5.2.3

Move

- 6

$- 6$ to the left of

a

$a$ .

8 - 6 \cdot a + a \cdot a

$8 - 6 \cdot a + a \cdot a$

Step 5.2.4

Multiply

a

$a$ by

a

$a$ .

8 - 6 a + a^{2}

$8 - 6 a + a^{2}$

8 - 6 a + a^{2}

$8 - 6 a + a^{2}$

8 - 6 a + a^{2}

$8 - 6 a + a^{2}$