Linear Algebra Examples

$[\begin{array}{c} a & b & 0 & 0 \\ c & d & 0 & 0 \\ 0 & 0 & a & - b \\ 0 & 0 & c & d \end{array}]$

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.

$| \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Step 1.4

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

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Step 1.5

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

$| \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Step 1.6

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

$- b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$

Step 1.7

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

$| \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} |$

Step 1.8

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

$0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} |$

Step 1.9

The minor for $a_{14}$ is the determinant with row $1$ and column $4$ deleted.

$| \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Step 1.10

Multiply element $a_{14}$ by its cofactor.

$0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Step 1.11

Add the terms together.

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} | + 0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Step 2

Multiply $0$ by $| \begin{array}{c} c & d & 0 \\ 0 & 0 & - b \\ 0 & 0 & d \end{array} |$ .

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0 | \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$

Step 3

Multiply $0$ by $| \begin{array}{c} c & d & 0 \\ 0 & 0 & a \\ 0 & 0 & c \end{array} |$ .

$a | \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Step 4

Evaluate $| \begin{array}{c} d & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$ .

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Step 4.1

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 4.1.1

Consider the corresponding sign chart.

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

Step 4.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

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

$| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$

Step 4.1.6

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

$0 | \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$

Step 4.1.7

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

$| \begin{array}{c} 0 & a \\ 0 & c \end{array} |$

Step 4.1.8

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

$0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |$

Step 4.1.9

Add the terms together.

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

Step 4.2

Multiply $0$ by $| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$ .

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

Step 4.3

Multiply $0$ by $| \begin{array}{c} 0 & a \\ 0 & c \end{array} |$ .

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Simplify each term.

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Step 4.4.2.1

Rewrite using the commutative property of multiplication.

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

Step 4.4.2.2

Multiply $- 1$ by $- 1$ .

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

Step 4.4.2.3

Multiply $c$ by $1$ .

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

Step 4.5

Simplify the determinant.

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Step 4.5.1

Combine the opposite terms in $d (a d + c b) + 0 + 0$ .

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Step 4.5.1.1

Add $d (a d + c b)$ and $0$ .

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

Step 4.5.1.2

Add $d (a d + c b)$ and $0$ .

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

Step 4.5.2

Apply the distributive property.

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

Step 4.5.3

Multiply $d$ by $d$ by adding the exponents.

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Step 4.5.3.1

Move $d$ .

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

Step 4.5.3.2

Multiply $d$ by $d$ .

$a (d^{2} a + d (c b)) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

$a (d^{2} a + d c b) - b | \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} | + 0 + 0$

Step 5

Evaluate $| \begin{array}{c} c & 0 & 0 \\ 0 & a & - b \\ 0 & c & d \end{array} |$ .

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Step 5.1

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 5.1.1

Consider the corresponding sign chart.

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

Step 5.1.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 5.1.3

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

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

Step 5.1.4

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

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

Step 5.1.5

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

$| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$

Step 5.1.6

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

$0 | \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$

Step 5.1.7

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

$| \begin{array}{c} 0 & a \\ 0 & c \end{array} |$

Step 5.1.8

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

$0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |$

Step 5.1.9

Add the terms together.

$a (d^{2} a + d c b) - b (c | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 | \begin{array}{c} 0 & - b \\ 0 & d \end{array} | + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) + 0 + 0$

Step 5.2

Multiply $0$ by $| \begin{array}{c} 0 & - b \\ 0 & d \end{array} |$ .

$a (d^{2} a + d c b) - b (c | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0 | \begin{array}{c} 0 & a \\ 0 & c \end{array} |) + 0 + 0$

Step 5.3

Multiply $0$ by $| \begin{array}{c} 0 & a \\ 0 & c \end{array} |$ .

$a (d^{2} a + d c b) - b (c | \begin{array}{c} a & - b \\ c & d \end{array} | + 0 + 0) + 0 + 0$

Step 5.4

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

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Step 5.4.1

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

$a (d^{2} a + d c b) - b (c (a d - c (- b)) + 0 + 0) + 0 + 0$

Step 5.4.2

Simplify each term.

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Step 5.4.2.1

Rewrite using the commutative property of multiplication.

$a (d^{2} a + d c b) - b (c (a d - 1 \cdot - 1 c b) + 0 + 0) + 0 + 0$

Step 5.4.2.2

Multiply $- 1$ by $- 1$ .

$a (d^{2} a + d c b) - b (c (a d + 1 c b) + 0 + 0) + 0 + 0$

Step 5.4.2.3

Multiply $c$ by $1$ .

$a (d^{2} a + d c b) - b (c (a d + c b) + 0 + 0) + 0 + 0$

Step 5.5

Simplify the determinant.

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Step 5.5.1

Combine the opposite terms in $c (a d + c b) + 0 + 0$ .

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Step 5.5.1.1

Add $c (a d + c b)$ and $0$ .

$a (d^{2} a + d c b) - b (c (a d + c b) + 0) + 0 + 0$

Step 5.5.1.2

Add $c (a d + c b)$ and $0$ .

$a (d^{2} a + d c b) - b (c (a d + c b)) + 0 + 0$

Step 5.5.2

Apply the distributive property.

$a (d^{2} a + d c b) - b (c (a d) + c (c b)) + 0 + 0$

Step 5.5.3

Multiply $c$ by $c$ by adding the exponents.

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Step 5.5.3.1

Move $c$ .

$a (d^{2} a + d c b) - b (c a d + c \cdot c b) + 0 + 0$

Step 5.5.3.2

Multiply $c$ by $c$ .

$a (d^{2} a + d c b) - b (c a d + c^{2} b) + 0 + 0$

Step 6

Simplify the determinant.

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Step 6.1

Combine the opposite terms in $a (d^{2} a + d c b) - b (c a d + c^{2} b) + 0 + 0$ .

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Step 6.1.1

Add $a (d^{2} a + d c b) - b (c a d + c^{2} b)$ and $0$ .

$a (d^{2} a + d c b) - b (c a d + c^{2} b) + 0$

Step 6.1.2

Add $a (d^{2} a + d c b) - b (c a d + c^{2} b)$ and $0$ .

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

Step 6.2

Simplify each term.

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Step 6.2.1

Apply the distributive property.

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

Step 6.2.2

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

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Step 6.2.2.1

Move $a$ .

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

Step 6.2.2.2

Multiply $a$ by $a$ .

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

Step 6.2.3

Apply the distributive property.

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

Step 6.2.4

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

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Step 6.2.4.1

Move $b$ .

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

Step 6.2.4.2

Multiply $b$ by $b$ .

$a^{2} d^{2} + a d c b - b c a d - b^{2} c^{2}$

Step 6.3

Combine the opposite terms in $a^{2} d^{2} + a d c b - b c a d - b^{2} c^{2}$ .

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Step 6.3.1

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

$a^{2} d^{2} + a b c d - a b c d - b^{2} c^{2}$

Step 6.3.2

Subtract $a b c d$ from $a b c d$ .

$a^{2} d^{2} + 0 - b^{2} c^{2}$

Step 6.3.3

Add $a^{2} d^{2}$ and $0$ .

$a^{2} d^{2} - b^{2} c^{2}$