Finite Math Examples

$[\begin{array}{c} 1 & 1 & 1 \\ x & y & z \\ z y & x z & x y \end{array}]$

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

$| \begin{array}{c} 1 & 1 \\ x z & x y \end{array} |$

Step 1.4

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

$- x | \begin{array}{c} 1 & 1 \\ x z & x y \end{array} |$

Step 1.5

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

$| \begin{array}{c} 1 & 1 \\ z y & x y \end{array} |$

Step 1.6

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

$y | \begin{array}{c} 1 & 1 \\ z y & x y \end{array} |$

Step 1.7

The minor for $a_{23}$ is the determinant with row $2$ and column $3$ deleted.

$| \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$

Step 1.8

Multiply element $a_{23}$ by its cofactor.

$- z | \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$

Step 1.9

Add the terms together.

$- x | \begin{array}{c} 1 & 1 \\ x z & x y \end{array} | + y | \begin{array}{c} 1 & 1 \\ z y & x y \end{array} | - z | \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$

Step 2

Evaluate $| \begin{array}{c} 1 & 1 \\ x z & x y \end{array} |$ .

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Step 2.1

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

$- x (1 (x y) - x z \cdot 1) + y | \begin{array}{c} 1 & 1 \\ z y & x y \end{array} | - z | \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x y$ by $1$ .

$- x (x y - x z \cdot 1) + y | \begin{array}{c} 1 & 1 \\ z y & x y \end{array} | - z | \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$

Step 2.2.2

Multiply $- 1$ by $1$ .

$- x (x y - x z) + y | \begin{array}{c} 1 & 1 \\ z y & x y \end{array} | - z | \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 1 & 1 \\ z y & x y \end{array} |$ .

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Step 3.1

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

$- x (x y - x z) + y (1 (x y) - z y \cdot 1) - z | \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $x y$ by $1$ .

$- x (x y - x z) + y (x y - z y \cdot 1) - z | \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$

Step 3.2.2

Multiply $- 1$ by $1$ .

$- x (x y - x z) + y (x y - z y) - z | \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 1 & 1 \\ z y & x z \end{array} |$ .

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Step 4.1

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

$- x (x y - x z) + y (x y - z y) - z (1 (x z) - z y \cdot 1)$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply $x z$ by $1$ .

$- x (x y - x z) + y (x y - z y) - z (x z - z y \cdot 1)$

Step 4.2.2

Multiply $- 1$ by $1$ .

$- x (x y - x z) + y (x y - z y) - z (x z - z y)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$- x (x y) - x (- x z) + y (x y - z y) - z (x z - z y)$

Step 5.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.2.1

Move $x$ .

$- (x \cdot x) y - x (- x z) + y (x y - z y) - z (x z - z y)$

Step 5.2.2

Multiply $x$ by $x$ .

$- x^{2} y - x (- x z) + y (x y - z y) - z (x z - z y)$

Step 5.3

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1

Move $x$ .

$- x^{2} y - (x \cdot x) (- 1 z) + y (x y - z y) - z (x z - z y)$

Step 5.3.2

Multiply $x$ by $x$ .

$- x^{2} y - x^{2} (- 1 z) + y (x y - z y) - z (x z - z y)$

Step 5.4

Simplify each term.

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Step 5.4.1

Rewrite using the commutative property of multiplication.

$- x^{2} y - 1 \cdot - 1 x^{2} z + y (x y - z y) - z (x z - z y)$

Step 5.4.2

Multiply $- 1$ by $- 1$ .

$- x^{2} y + 1 x^{2} z + y (x y - z y) - z (x z - z y)$

Step 5.4.3

Multiply $x^{2}$ by $1$ .

$- x^{2} y + x^{2} z + y (x y - z y) - z (x z - z y)$

Step 5.5

Apply the distributive property.

$- x^{2} y + x^{2} z + y (x y) + y (- z y) - z (x z - z y)$

Step 5.6

Multiply $y$ by $y$ by adding the exponents.

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Step 5.6.1

Move $y$ .

$- x^{2} y + x^{2} z + y \cdot y x + y (- z y) - z (x z - z y)$

Step 5.6.2

Multiply $y$ by $y$ .

$- x^{2} y + x^{2} z + y^{2} x + y (- z y) - z (x z - z y)$

Step 5.7

Rewrite using the commutative property of multiplication.

$- x^{2} y + x^{2} z + y^{2} x - y (z y) - z (x z - z y)$

Step 5.8

Multiply $y$ by $y$ by adding the exponents.

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Step 5.8.1

Move $y$ .

$- x^{2} y + x^{2} z + y^{2} x - (y \cdot y) z - z (x z - z y)$

Step 5.8.2

Multiply $y$ by $y$ .

$- x^{2} y + x^{2} z + y^{2} x - y^{2} z - z (x z - z y)$

Step 5.9

Apply the distributive property.

$- x^{2} y + x^{2} z + y^{2} x - y^{2} z - z (x z) - z (- z y)$

Step 5.10

Multiply $z$ by $z$ by adding the exponents.

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Step 5.10.1

Move $z$ .

$- x^{2} y + x^{2} z + y^{2} x - y^{2} z - (z \cdot z) x - z (- z y)$

Step 5.10.2

Multiply $z$ by $z$ .

$- x^{2} y + x^{2} z + y^{2} x - y^{2} z - z^{2} x - z (- z y)$

Step 5.11

Multiply $z$ by $z$ by adding the exponents.

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Step 5.11.1

Move $z$ .

$- x^{2} y + x^{2} z + y^{2} x - y^{2} z - z^{2} x - (z \cdot z) (- 1 y)$

Step 5.11.2

Multiply $z$ by $z$ .

$- x^{2} y + x^{2} z + y^{2} x - y^{2} z - z^{2} x - z^{2} (- 1 y)$

Step 5.12

Simplify each term.

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Step 5.12.1

Rewrite using the commutative property of multiplication.

$- x^{2} y + x^{2} z + y^{2} x - y^{2} z - z^{2} x - 1 \cdot - 1 z^{2} y$

Step 5.12.2

Multiply $- 1$ by $- 1$ .

$- x^{2} y + x^{2} z + y^{2} x - y^{2} z - z^{2} x + 1 z^{2} y$

Step 5.12.3

Multiply $z^{2}$ by $1$ .

$- x^{2} y + x^{2} z + y^{2} x - y^{2} z - z^{2} x + z^{2} y$