Linear Algebra Examples

$[\begin{array}{c} 1 & x & x^{2} \\ 1 & x^{2} & x^{4} \\ 1 & x^{4} & x^{6} \end{array}]$

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 column $2$ by its cofactor and add.

Tap for more steps...

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

$| \begin{array}{c} 1 & x^{4} \\ 1 & x^{6} \end{array} |$

Step 1.4

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

$- x | \begin{array}{c} 1 & x^{4} \\ 1 & x^{6} \end{array} |$

Step 1.5

The minor for $a_{22}$ is the determinant with row $2$ and column $2$ deleted.

$| \begin{array}{c} 1 & x^{2} \\ 1 & x^{6} \end{array} |$

Step 1.6

Multiply element $a_{22}$ by its cofactor.

$x^{2} | \begin{array}{c} 1 & x^{2} \\ 1 & x^{6} \end{array} |$

Step 1.7

The minor for $a_{32}$ is the determinant with row $3$ and column $2$ deleted.

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

Step 1.8

Multiply element $a_{32}$ by its cofactor.

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

Step 1.9

Add the terms together.

$- x | \begin{array}{c} 1 & x^{4} \\ 1 & x^{6} \end{array} | + x^{2} | \begin{array}{c} 1 & x^{2} \\ 1 & x^{6} \end{array} | - x^{4} | \begin{array}{c} 1 & x^{2} \\ 1 & x^{4} \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 1 & x^{4} \\ 1 & x^{6} \end{array} |$ .

Tap for more steps...

Step 2.1

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

$- x (1 x^{6} - x^{4}) + x^{2} | \begin{array}{c} 1 & x^{2} \\ 1 & x^{6} \end{array} | - x^{4} | \begin{array}{c} 1 & x^{2} \\ 1 & x^{4} \end{array} |$

Step 2.2

Multiply $x^{6}$ by $1$ .

$- x (x^{6} - x^{4}) + x^{2} | \begin{array}{c} 1 & x^{2} \\ 1 & x^{6} \end{array} | - x^{4} | \begin{array}{c} 1 & x^{2} \\ 1 & x^{4} \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 1 & x^{2} \\ 1 & x^{6} \end{array} |$ .

Tap for more steps...

Step 3.1

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

$- x (x^{6} - x^{4}) + x^{2} (1 x^{6} - x^{2}) - x^{4} | \begin{array}{c} 1 & x^{2} \\ 1 & x^{4} \end{array} |$

Step 3.2

Multiply $x^{6}$ by $1$ .

$- x (x^{6} - x^{4}) + x^{2} (x^{6} - x^{2}) - x^{4} | \begin{array}{c} 1 & x^{2} \\ 1 & x^{4} \end{array} |$

Step 4

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

Tap for more steps...

Step 4.1

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

$- x (x^{6} - x^{4}) + x^{2} (x^{6} - x^{2}) - x^{4} (1 x^{4} - x^{2})$

Step 4.2

Multiply $x^{4}$ by $1$ .

$- x (x^{6} - x^{4}) + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5

Simplify the determinant.

Tap for more steps...

Step 5.1

Simplify each term.

Tap for more steps...

Step 5.1.1

Apply the distributive property.

$- x \cdot x^{6} - x (- x^{4}) + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.2

Multiply $x$ by $x^{6}$ by adding the exponents.

Tap for more steps...

Step 5.1.2.1

Move $x^{6}$ .

$- (x^{6} x) - x (- x^{4}) + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.2.2

Multiply $x^{6}$ by $x$ .

Tap for more steps...

Step 5.1.2.2.1

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

$- (x^{6} x^{1}) - x (- x^{4}) + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.2.2.2

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

$- x^{6 + 1} - x (- x^{4}) + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.2.3

Add $6$ and $1$ .

$- x^{7} - x (- x^{4}) + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.3

Rewrite using the commutative property of multiplication.

$- x^{7} - 1 \cdot - 1 x \cdot x^{4} + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.4

Simplify each term.

Tap for more steps...

Step 5.1.4.1

Multiply $x$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 5.1.4.1.1

Move $x^{4}$ .

$- x^{7} - 1 \cdot - 1 (x^{4} x) + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.4.1.2

Multiply $x^{4}$ by $x$ .

Tap for more steps...

Step 5.1.4.1.2.1

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

$- x^{7} - 1 \cdot - 1 (x^{4} x^{1}) + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.4.1.2.2

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

$- x^{7} - 1 \cdot - 1 x^{4 + 1} + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.4.1.3

Add $4$ and $1$ .

$- x^{7} - 1 \cdot - 1 x^{5} + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.4.2

Multiply $- 1$ by $- 1$ .

$- x^{7} + 1 x^{5} + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.4.3

Multiply $x^{5}$ by $1$ .

$- x^{7} + x^{5} + x^{2} (x^{6} - x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.5

Apply the distributive property.

$- x^{7} + x^{5} + x^{2} x^{6} + x^{2} (- x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.6

Multiply $x^{2}$ by $x^{6}$ by adding the exponents.

Tap for more steps...

Step 5.1.6.1

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

$- x^{7} + x^{5} + x^{2 + 6} + x^{2} (- x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.6.2

Add $2$ and $6$ .

$- x^{7} + x^{5} + x^{8} + x^{2} (- x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.7

Rewrite using the commutative property of multiplication.

$- x^{7} + x^{5} + x^{8} - x^{2} x^{2} - x^{4} (x^{4} - x^{2})$

Step 5.1.8

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

Tap for more steps...

Step 5.1.8.1

Move $x^{2}$ .

$- x^{7} + x^{5} + x^{8} - (x^{2} x^{2}) - x^{4} (x^{4} - x^{2})$

Step 5.1.8.2

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

$- x^{7} + x^{5} + x^{8} - x^{2 + 2} - x^{4} (x^{4} - x^{2})$

Step 5.1.8.3

Add $2$ and $2$ .

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{4} (x^{4} - x^{2})$

Step 5.1.9

Apply the distributive property.

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{4} x^{4} - x^{4} (- x^{2})$

Step 5.1.10

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

Tap for more steps...

Step 5.1.10.1

Move $x^{4}$ .

$- x^{7} + x^{5} + x^{8} - x^{4} - (x^{4} x^{4}) - x^{4} (- x^{2})$

Step 5.1.10.2

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

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{4 + 4} - x^{4} (- x^{2})$

Step 5.1.10.3

Add $4$ and $4$ .

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{8} - x^{4} (- x^{2})$

Step 5.1.11

Rewrite using the commutative property of multiplication.

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{8} - 1 \cdot - 1 x^{4} x^{2}$

Step 5.1.12

Simplify each term.

Tap for more steps...

Step 5.1.12.1

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 5.1.12.1.1

Move $x^{2}$ .

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{8} - 1 \cdot - 1 (x^{2} x^{4})$

Step 5.1.12.1.2

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

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{8} - 1 \cdot - 1 x^{2 + 4}$

Step 5.1.12.1.3

Add $2$ and $4$ .

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{8} - 1 \cdot - 1 x^{6}$

Step 5.1.12.2

Multiply $- 1$ by $- 1$ .

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{8} + 1 x^{6}$

Step 5.1.12.3

Multiply $x^{6}$ by $1$ .

$- x^{7} + x^{5} + x^{8} - x^{4} - x^{8} + x^{6}$

Step 5.2

Combine the opposite terms in $- x^{7} + x^{5} + x^{8} - x^{4} - x^{8} + x^{6}$ .

Tap for more steps...

Step 5.2.1

Subtract $x^{8}$ from $x^{8}$ .

$- x^{7} + x^{5} - x^{4} + 0 + x^{6}$

Step 5.2.2

Add $- x^{7} + x^{5} - x^{4}$ and $0$ .

$- x^{7} + x^{5} - x^{4} + x^{6}$