Linear Algebra Examples

$[\begin{array}{c} 1 & a & b + c \\ 1 & b & a + c \\ 1 & c & a + b \end{array}]$

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.

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

$| \begin{array}{c} 1 & a + c \\ 1 & a + b \end{array} |$

Step 1.4

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

$- a | \begin{array}{c} 1 & a + c \\ 1 & a + b \end{array} |$

Step 1.5

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

$| \begin{array}{c} 1 & b + c \\ 1 & a + b \end{array} |$

Step 1.6

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

$b | \begin{array}{c} 1 & b + c \\ 1 & a + b \end{array} |$

Step 1.7

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

$| \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 1.8

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

$- c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 1.9

Add the terms together.

$- a | \begin{array}{c} 1 & a + c \\ 1 & a + b \end{array} | + b | \begin{array}{c} 1 & b + c \\ 1 & a + b \end{array} | - c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 1 & a + c \\ 1 & a + b \end{array} |$ .

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Step 2.1

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

$- a (1 (a + b) - (a + c)) + b | \begin{array}{c} 1 & b + c \\ 1 & a + b \end{array} | - c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $a + b$ by $1$ .

$- a (a + b - (a + c)) + b | \begin{array}{c} 1 & b + c \\ 1 & a + b \end{array} | - c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 2.2.1.2

Apply the distributive property.

$- a (a + b - a - c) + b | \begin{array}{c} 1 & b + c \\ 1 & a + b \end{array} | - c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 2.2.2

Combine the opposite terms in $a + b - a - c$ .

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Step 2.2.2.1

Subtract $a$ from $a$ .

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

Step 2.2.2.2

Add $b$ and $0$ .

$- a (b - c) + b | \begin{array}{c} 1 & b + c \\ 1 & a + b \end{array} | - c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 1 & b + c \\ 1 & a + b \end{array} |$ .

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Step 3.1

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

$- a (b - c) + b (1 (a + b) - (b + c)) - c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

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 $a + b$ by $1$ .

$- a (b - c) + b (a + b - (b + c)) - c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 3.2.1.2

Apply the distributive property.

$- a (b - c) + b (a + b - b - c) - c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 3.2.2

Combine the opposite terms in $a + b - b - c$ .

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Step 3.2.2.1

Subtract $b$ from $b$ .

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

Step 3.2.2.2

Add $a$ and $0$ .

$- a (b - c) + b (a - c) - c | \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 1 & b + c \\ 1 & a + c \end{array} |$ .

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Step 4.1

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

$- a (b - c) + b (a - c) - c (1 (a + c) - (b + c))$

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 $a + c$ by $1$ .

$- a (b - c) + b (a - c) - c (a + c - (b + c))$

Step 4.2.1.2

Apply the distributive property.

$- a (b - c) + b (a - c) - c (a + c - b - c)$

Step 4.2.2

Combine the opposite terms in $a + c - b - c$ .

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Step 4.2.2.1

Subtract $c$ from $c$ .

$- a (b - c) + b (a - c) - c (a - b + 0)$

Step 4.2.2.2

Add $a - b$ and $0$ .

$- a (b - c) + b (a - c) - c (a - b)$

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.

$- a b - a (- c) + b (a - c) - c (a - b)$

Step 5.1.2

Rewrite using the commutative property of multiplication.

$- a b - 1 \cdot - 1 a c + b (a - c) - c (a - b)$

Step 5.1.3

Simplify each term.

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Step 5.1.3.1

Multiply $- 1$ by $- 1$ .

$- a b + 1 a c + b (a - c) - c (a - b)$

Step 5.1.3.2

Multiply $a$ by $1$ .

$- a b + a c + b (a - c) - c (a - b)$

Step 5.1.4

Apply the distributive property.

$- a b + a c + b a + b (- c) - c (a - b)$

Step 5.1.5

Rewrite using the commutative property of multiplication.

$- a b + a c + b a - b c - c (a - b)$

Step 5.1.6

Apply the distributive property.

$- a b + a c + b a - b c - c a - c (- b)$

Step 5.1.7

Rewrite using the commutative property of multiplication.

$- a b + a c + b a - b c - c a - 1 \cdot - 1 c b$

Step 5.1.8

Simplify each term.

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Step 5.1.8.1

Multiply $- 1$ by $- 1$ .

$- a b + a c + b a - b c - c a + 1 c b$

Step 5.1.8.2

Multiply $c$ by $1$ .

$- a b + a c + b a - b c - c a + c b$

Step 5.2

Combine the opposite terms in $- a b + a c + b a - b c - c a + c b$ .

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Step 5.2.1

Reorder the factors in the terms $- a b$ and $b a$ .

$- a b + a c + a b - b c - c a + c b$

Step 5.2.2

Add $- a b$ and $a b$ .

$a c + 0 - b c - c a + c b$

Step 5.2.3

Add $a c$ and $0$ .

$a c - b c - c a + c b$

Step 5.2.4

Reorder the factors in the terms $a c$ and $- c a$ .

$a c - b c - a c + c b$

Step 5.2.5

Subtract $a c$ from $a c$ .

$- b c + 0 + c b$

Step 5.2.6

Add $- b c$ and $0$ .

$- b c + c b$

Step 5.2.7

Reorder the factors in the terms $- b c$ and $c b$ .

$- b c + b c$

Step 5.2.8

Add $- b c$ and $b c$ .

$0$