Linear Algebra Examples

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

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 $1$ 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.

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

Step 1.4

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

$(2 q - 1) | \begin{array}{c} 1 & q \\ 1 & 2 q - 1 \end{array} |$

Step 1.5

The minor for $a_{21}$ is the determinant with row $2$ and column $1$ deleted.

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

Step 1.6

Multiply element $a_{21}$ by its cofactor.

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

Step 1.7

The minor for $a_{31}$ is the determinant with row $3$ and column $1$ deleted.

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

Step 1.8

Multiply element $a_{31}$ by its cofactor.

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

Step 1.9

Add the terms together.

$(2 q - 1) | \begin{array}{c} 1 & q \\ 1 & 2 q - 1 \end{array} | - q | \begin{array}{c} 1 & 1 \\ 1 & 2 q - 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & q \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 1 & q \\ 1 & 2 q - 1 \end{array} |$ .

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Step 2.1

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

$(2 q - 1) (1 (2 q - 1) - q) - q | \begin{array}{c} 1 & 1 \\ 1 & 2 q - 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & q \end{array} |$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Multiply $2 q - 1$ by $1$ .

$(2 q - 1) (2 q - 1 - q) - q | \begin{array}{c} 1 & 1 \\ 1 & 2 q - 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & q \end{array} |$

Step 2.2.2

Subtract $q$ from $2 q$ .

$(2 q - 1) (q - 1) - q | \begin{array}{c} 1 & 1 \\ 1 & 2 q - 1 \end{array} | + 1 | \begin{array}{c} 1 & 1 \\ 1 & q \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 1 & 1 \\ 1 & 2 q - 1 \end{array} |$ .

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Step 3.1

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

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

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 $2 q - 1$ by $1$ .

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

Step 3.2.1.2

Multiply $- 1$ by $1$ .

$(2 q - 1) (q - 1) - q (2 q - 1 - 1) + 1 | \begin{array}{c} 1 & 1 \\ 1 & q \end{array} |$

Step 3.2.2

Subtract $1$ from $- 1$ .

$(2 q - 1) (q - 1) - q (2 q - 2) + 1 | \begin{array}{c} 1 & 1 \\ 1 & q \end{array} |$

Step 4

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

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Step 4.1

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

$(2 q - 1) (q - 1) - q (2 q - 2) + 1 (1 q - 1 \cdot 1)$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $q$ by $1$ .

$(2 q - 1) (q - 1) - q (2 q - 2) + 1 (q - 1 \cdot 1)$

Step 4.2.2

Multiply $- 1$ by $1$ .

$(2 q - 1) (q - 1) - q (2 q - 2) + 1 (q - 1)$

Step 5

Simplify the determinant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Expand $(2 q - 1) (q - 1)$ using the FOIL Method.

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Step 5.1.1.1

Apply the distributive property.

$2 q (q - 1) - 1 (q - 1) - q (2 q - 2) + 1 (q - 1)$

Step 5.1.1.2

Apply the distributive property.

$2 q \cdot q + 2 q \cdot - 1 - 1 (q - 1) - q (2 q - 2) + 1 (q - 1)$

Step 5.1.1.3

Apply the distributive property.

$2 q \cdot q + 2 q \cdot - 1 - 1 q - 1 \cdot - 1 - q (2 q - 2) + 1 (q - 1)$

Step 5.1.2

Simplify and combine like terms.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Multiply $q$ by $q$ by adding the exponents.

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Step 5.1.2.1.1.1

Move $q$ .

$2 (q \cdot q) + 2 q \cdot - 1 - 1 q - 1 \cdot - 1 - q (2 q - 2) + 1 (q - 1)$

Step 5.1.2.1.1.2

Multiply $q$ by $q$ .

$2 q^{2} + 2 q \cdot - 1 - 1 q - 1 \cdot - 1 - q (2 q - 2) + 1 (q - 1)$

Step 5.1.2.1.2

Multiply $- 1$ by $2$ .

$2 q^{2} - 2 q - 1 q - 1 \cdot - 1 - q (2 q - 2) + 1 (q - 1)$

Step 5.1.2.1.3

Rewrite $- 1 q$ as $- q$ .

$2 q^{2} - 2 q - q - 1 \cdot - 1 - q (2 q - 2) + 1 (q - 1)$

Step 5.1.2.1.4

Multiply $- 1$ by $- 1$ .

$2 q^{2} - 2 q - q + 1 - q (2 q - 2) + 1 (q - 1)$

Step 5.1.2.2

Subtract $q$ from $- 2 q$ .

$2 q^{2} - 3 q + 1 - q (2 q - 2) + 1 (q - 1)$

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

Rewrite using the commutative property of multiplication.

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

Step 5.1.5

Multiply $- 2$ by $- 1$ .

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

Step 5.1.6

Simplify each term.

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Step 5.1.6.1

Multiply $q$ by $q$ by adding the exponents.

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Step 5.1.6.1.1

Move $q$ .

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

Step 5.1.6.1.2

Multiply $q$ by $q$ .

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

Step 5.1.6.2

Multiply $- 1$ by $2$ .

$2 q^{2} - 3 q + 1 - 2 q^{2} + 2 q + 1 (q - 1)$

Step 5.1.7

Multiply $q - 1$ by $1$ .

$2 q^{2} - 3 q + 1 - 2 q^{2} + 2 q + q - 1$

Step 5.2

Combine the opposite terms in $2 q^{2} - 3 q + 1 - 2 q^{2} + 2 q + q - 1$ .

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Step 5.2.1

Subtract $2 q^{2}$ from $2 q^{2}$ .

$- 3 q + 1 + 0 + 2 q + q - 1$

Step 5.2.2

Add $- 3 q + 1$ and $0$ .

$- 3 q + 1 + 2 q + q - 1$

Step 5.2.3

Subtract $1$ from $1$ .

$- 3 q + 2 q + q + 0$

Step 5.2.4

Add $- 3 q + 2 q + q$ and $0$ .

$- 3 q + 2 q + q$

Step 5.3

Add $- 3 q$ and $2 q$ .

$- q + q$

Step 5.4

Add $- q$ and $q$ .

$0$