Finite Math Examples

$[\begin{array}{c} 4 - x & 2 \\ - 5 & - 2 - x \end{array}]$

Step 1

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

$(4 - x) (- 2 - x) - (- 5 \cdot 2)$

Step 2

Simplify the determinant.

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Step 2.1

Simplify each term.

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Step 2.1.1

Expand $(4 - x) (- 2 - x)$ using the FOIL Method.

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Step 2.1.1.1

Apply the distributive property.

$4 (- 2 - x) - x (- 2 - x) - (- 5 \cdot 2)$

Step 2.1.1.2

Apply the distributive property.

$4 \cdot - 2 + 4 (- x) - x (- 2 - x) - (- 5 \cdot 2)$

Step 2.1.1.3

Apply the distributive property.

$4 \cdot - 2 + 4 (- x) - x \cdot - 2 - x (- x) - (- 5 \cdot 2)$

Step 2.1.2

Simplify and combine like terms.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Multiply $4$ by $- 2$ .

$- 8 + 4 (- x) - x \cdot - 2 - x (- x) - (- 5 \cdot 2)$

Step 2.1.2.1.2

Multiply $- 1$ by $4$ .

$- 8 - 4 x - x \cdot - 2 - x (- x) - (- 5 \cdot 2)$

Step 2.1.2.1.3

Multiply $- 2$ by $- 1$ .

$- 8 - 4 x + 2 x - x (- x) - (- 5 \cdot 2)$

Step 2.1.2.1.4

Rewrite using the commutative property of multiplication.

$- 8 - 4 x + 2 x - 1 \cdot - 1 x \cdot x - (- 5 \cdot 2)$

Step 2.1.2.1.5

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

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Step 2.1.2.1.5.1

Move $x$ .

$- 8 - 4 x + 2 x - 1 \cdot - 1 (x \cdot x) - (- 5 \cdot 2)$

Step 2.1.2.1.5.2

Multiply $x$ by $x$ .

$- 8 - 4 x + 2 x - 1 \cdot - 1 x^{2} - (- 5 \cdot 2)$

Step 2.1.2.1.6

Multiply $- 1$ by $- 1$ .

$- 8 - 4 x + 2 x + 1 x^{2} - (- 5 \cdot 2)$

Step 2.1.2.1.7

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

$- 8 - 4 x + 2 x + x^{2} - (- 5 \cdot 2)$

Step 2.1.2.2

Add $- 4 x$ and $2 x$ .

$- 8 - 2 x + x^{2} - (- 5 \cdot 2)$

Step 2.1.3

Multiply $- (- 5 \cdot 2)$ .

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Step 2.1.3.1

Multiply $- 5$ by $2$ .

$- 8 - 2 x + x^{2} - - 10$

Step 2.1.3.2

Multiply $- 1$ by $- 10$ .

$- 8 - 2 x + x^{2} + 10$

Step 2.2

Add $- 8$ and $10$ .

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