Linear Algebra Examples

$[\begin{array}{c} a & a & b \\ a & b & a \\ b & a & a \end{array}]$

Step 1

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 1.1

Consider the corresponding sign chart.

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

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 $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} b & a \\ a & a \end{array} |$

Step 1.4

Multiply element $a_{11}$ by its cofactor.

$a | \begin{array}{c} b & a \\ a & a \end{array} |$

Step 1.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} a & a \\ b & a \end{array} |$

Step 1.6

Multiply element $a_{12}$ by its cofactor.

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

Step 1.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} a & b \\ b & a \end{array} |$

Step 1.8

Multiply element $a_{13}$ by its cofactor.

$b | \begin{array}{c} a & b \\ b & a \end{array} |$

Step 1.9

Add the terms together.

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

Step 2

Evaluate $| \begin{array}{c} b & a \\ a & a \end{array} |$ .

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Step 2.1

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

$a (b a - a \cdot a) - a | \begin{array}{c} a & a \\ b & a \end{array} | + b | \begin{array}{c} a & b \\ b & a \end{array} |$

Step 2.2

Multiply $a$ by $a$ by adding the exponents.

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Step 2.2.1

Move $a$ .

$a (b a - (a \cdot a)) - a | \begin{array}{c} a & a \\ b & a \end{array} | + b | \begin{array}{c} a & b \\ b & a \end{array} |$

Step 2.2.2

Multiply $a$ by $a$ .

$a (b a - a^{2}) - a | \begin{array}{c} a & a \\ b & a \end{array} | + b | \begin{array}{c} a & b \\ b & a \end{array} |$

Step 3

Evaluate $| \begin{array}{c} a & a \\ b & a \end{array} |$ .

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Step 3.1

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

$a (b a - a^{2}) - a (a \cdot a - b a) + b | \begin{array}{c} a & b \\ b & a \end{array} |$

Step 3.2

Multiply $a$ by $a$ .

$a (b a - a^{2}) - a (a^{2} - b a) + b | \begin{array}{c} a & b \\ b & a \end{array} |$

Step 4

Evaluate $| \begin{array}{c} a & b \\ b & a \end{array} |$ .

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Step 4.1

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

$a (b a - a^{2}) - a (a^{2} - b a) + b (a \cdot a - b \cdot b)$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $a$ by $a$ .

$a (b a - a^{2}) - a (a^{2} - b a) + b (a^{2} - b \cdot b)$

Step 4.2.2

Multiply $b$ by $b$ by adding the exponents.

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Step 4.2.2.1

Move $b$ .

$a (b a - a^{2}) - a (a^{2} - b a) + b (a^{2} - (b \cdot b))$

Step 4.2.2.2

Multiply $b$ by $b$ .

$a (b a - a^{2}) - a (a^{2} - b a) + b (a^{2} - b^{2})$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

$a (b a) + a (- a^{2}) - a (a^{2} - b a) + b (a^{2} - b^{2})$

Step 5.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 5.1.2.1

Move $a$ .

$a \cdot a b + a (- a^{2}) - a (a^{2} - b a) + b (a^{2} - b^{2})$

Step 5.1.2.2

Multiply $a$ by $a$ .

$a^{2} b + a (- a^{2}) - a (a^{2} - b a) + b (a^{2} - b^{2})$

Step 5.1.3

Rewrite using the commutative property of multiplication.

$a^{2} b - a \cdot a^{2} - a (a^{2} - b a) + b (a^{2} - b^{2})$

Step 5.1.4

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

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Step 5.1.4.1

Move $a^{2}$ .

$a^{2} b - (a^{2} a) - a (a^{2} - b a) + b (a^{2} - b^{2})$

Step 5.1.4.2

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

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Step 5.1.4.2.1

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

$a^{2} b - (a^{2} a^{1}) - a (a^{2} - b a) + b (a^{2} - b^{2})$

Step 5.1.4.2.2

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

$a^{2} b - a^{2 + 1} - a (a^{2} - b a) + b (a^{2} - b^{2})$

Step 5.1.4.3

Add $2$ and $1$ .

$a^{2} b - a^{3} - a (a^{2} - b a) + b (a^{2} - b^{2})$

Step 5.1.5

Apply the distributive property.

$a^{2} b - a^{3} - a \cdot a^{2} - a (- b a) + b (a^{2} - b^{2})$

Step 5.1.6

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

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Step 5.1.6.1

Move $a^{2}$ .

$a^{2} b - a^{3} - (a^{2} a) - a (- b a) + b (a^{2} - b^{2})$

Step 5.1.6.2

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

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Step 5.1.6.2.1

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

$a^{2} b - a^{3} - (a^{2} a^{1}) - a (- b a) + b (a^{2} - b^{2})$

Step 5.1.6.2.2

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

$a^{2} b - a^{3} - a^{2 + 1} - a (- b a) + b (a^{2} - b^{2})$

Step 5.1.6.3

Add $2$ and $1$ .

$a^{2} b - a^{3} - a^{3} - a (- b a) + b (a^{2} - b^{2})$

Step 5.1.7

Multiply $a$ by $a$ by adding the exponents.

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Step 5.1.7.1

Move $a$ .

$a^{2} b - a^{3} - a^{3} - (a \cdot a) (- b) + b (a^{2} - b^{2})$

Step 5.1.7.2

Multiply $a$ by $a$ .

$a^{2} b - a^{3} - a^{3} - a^{2} (- b) + b (a^{2} - b^{2})$

Step 5.1.8

Simplify each term.

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Step 5.1.8.1

Rewrite using the commutative property of multiplication.

$a^{2} b - a^{3} - a^{3} - 1 \cdot - 1 a^{2} b + b (a^{2} - b^{2})$

Step 5.1.8.2

Multiply $- 1$ by $- 1$ .

$a^{2} b - a^{3} - a^{3} + 1 a^{2} b + b (a^{2} - b^{2})$

Step 5.1.8.3

Multiply $a^{2}$ by $1$ .

$a^{2} b - a^{3} - a^{3} + a^{2} b + b (a^{2} - b^{2})$

Step 5.1.9

Apply the distributive property.

$a^{2} b - a^{3} - a^{3} + a^{2} b + b a^{2} + b (- b^{2})$

Step 5.1.10

Rewrite using the commutative property of multiplication.

$a^{2} b - a^{3} - a^{3} + a^{2} b + b a^{2} - b \cdot b^{2}$

Step 5.1.11

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

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Step 5.1.11.1

Move $b^{2}$ .

$a^{2} b - a^{3} - a^{3} + a^{2} b + b a^{2} - (b^{2} b)$

Step 5.1.11.2

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

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Step 5.1.11.2.1

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

$a^{2} b - a^{3} - a^{3} + a^{2} b + b a^{2} - (b^{2} b^{1})$

Step 5.1.11.2.2

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

$a^{2} b - a^{3} - a^{3} + a^{2} b + b a^{2} - b^{2 + 1}$

Step 5.1.11.3

Add $2$ and $1$ .

$a^{2} b - a^{3} - a^{3} + a^{2} b + b a^{2} - b^{3}$

Step 5.2

Add $a^{2} b$ and $a^{2} b$ .

$- a^{3} - a^{3} + 2 a^{2} b + b a^{2} - b^{3}$

Step 5.3

Subtract $a^{3}$ from $- a^{3}$ .

$- 2 a^{3} + 2 a^{2} b + b a^{2} - b^{3}$

Step 5.4

Add $2 a^{2} b$ and $b a^{2}$ .

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Step 5.4.1

Reorder $b$ and $a^{2}$ .

$- 2 a^{3} + 2 a^{2} b + a^{2} b - b^{3}$

Step 5.4.2

Add $2 a^{2} b$ and $a^{2} b$ .

$- 2 a^{3} + 3 a^{2} b - b^{3}$