Precalculus Examples

$[\begin{array}{c} x & y & z \\ u & v & w \\ - 3 & - 4 & 1 \end{array}]$

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 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.

$| \begin{array}{c} v & w \\ - 4 & 1 \end{array} |$

Step 1.4

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

$x | \begin{array}{c} v & w \\ - 4 & 1 \end{array} |$

Step 1.5

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

$| \begin{array}{c} u & w \\ - 3 & 1 \end{array} |$

Step 1.6

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

$- y | \begin{array}{c} u & w \\ - 3 & 1 \end{array} |$

Step 1.7

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

$| \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$

Step 1.8

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

$z | \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$

Step 1.9

Add the terms together.

$x | \begin{array}{c} v & w \\ - 4 & 1 \end{array} | - y | \begin{array}{c} u & w \\ - 3 & 1 \end{array} | + z | \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$

Step 2

Evaluate $| \begin{array}{c} v & w \\ - 4 & 1 \end{array} |$ .

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Step 2.1

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

$x (v \cdot 1 - (- 4 w)) - y | \begin{array}{c} u & w \\ - 3 & 1 \end{array} | + z | \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $v$ by $1$ .

$x (v - (- 4 w)) - y | \begin{array}{c} u & w \\ - 3 & 1 \end{array} | + z | \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$

Step 2.2.2

Multiply $- 4$ by $- 1$ .

$x (v + 4 w) - y | \begin{array}{c} u & w \\ - 3 & 1 \end{array} | + z | \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} u & w \\ - 3 & 1 \end{array} |$ .

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Step 3.1

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

$x (v + 4 w) - y (u \cdot 1 - (- 3 w)) + z | \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $u$ by $1$ .

$x (v + 4 w) - y (u - (- 3 w)) + z | \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$

Step 3.2.2

Multiply $- 3$ by $- 1$ .

$x (v + 4 w) - y (u + 3 w) + z | \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} u & v \\ - 3 & - 4 \end{array} |$ .

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Step 4.1

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

$x (v + 4 w) - y (u + 3 w) + z (u \cdot - 4 - (- 3 v))$

Step 4.2

Simplify each term.

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Step 4.2.1

Move $- 4$ to the left of $u$ .

$x (v + 4 w) - y (u + 3 w) + z (- 4 \cdot u - (- 3 v))$

Step 4.2.2

Multiply $- 3$ by $- 1$ .

$x (v + 4 w) - y (u + 3 w) + z (- 4 u + 3 v)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$x v + x (4 w) - y (u + 3 w) + z (- 4 u + 3 v)$

Step 5.2

Rewrite using the commutative property of multiplication.

$x v + 4 x w - y (u + 3 w) + z (- 4 u + 3 v)$

Step 5.3

Apply the distributive property.

$x v + 4 x w - y u - y (3 w) + z (- 4 u + 3 v)$

Step 5.4

Rewrite using the commutative property of multiplication.

$x v + 4 x w - y u - 1 \cdot 3 y w + z (- 4 u + 3 v)$

Step 5.5

Multiply $- 1$ by $3$ .

$x v + 4 x w - y u - 3 y w + z (- 4 u + 3 v)$

Step 5.6

Apply the distributive property.

$x v + 4 x w - y u - 3 y w + z (- 4 u) + z (3 v)$

Step 5.7

Rewrite using the commutative property of multiplication.

$x v + 4 x w - y u - 3 y w - 4 z u + z (3 v)$

Step 5.8

Rewrite using the commutative property of multiplication.

$x v + 4 x w - y u - 3 y w - 4 z u + 3 z v$