Linear Algebra Examples

$[\begin{array}{c} 3 & 2 h - 1 \\ h^{2} & - 49 \end{array}]$

Step 1

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

$3 \cdot - 49 - h^{2} (2 h - 1)$

Step 2

Simplify each term.

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Step 2.1

Multiply $3$ by $- 49$ .

$- 147 - h^{2} (2 h - 1)$

Step 2.2

Apply the distributive property.

$- 147 - h^{2} (2 h) - h^{2} \cdot - 1$

Step 2.3

Rewrite using the commutative property of multiplication.

$- 147 - 1 \cdot 2 h^{2} h - h^{2} \cdot - 1$

Step 2.4

Multiply $- h^{2} \cdot - 1$ .

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Step 2.4.1

Multiply $- 1$ by $- 1$ .

$- 147 - 1 \cdot 2 h^{2} h + 1 h^{2}$

Step 2.4.2

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

$- 147 - 1 \cdot 2 h^{2} h + h^{2}$

Step 2.5

Simplify each term.

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Step 2.5.1

Multiply $h^{2}$ by $h$ by adding the exponents.

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Step 2.5.1.1

Move $h$ .

$- 147 - 1 \cdot 2 (h \cdot h^{2}) + h^{2}$

Step 2.5.1.2

Multiply $h$ by $h^{2}$ .

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Step 2.5.1.2.1

Raise $h$ to the power of $1$ .

$- 147 - 1 \cdot 2 (h^{1} h^{2}) + h^{2}$

Step 2.5.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 147 - 1 \cdot 2 h^{1 + 2} + h^{2}$

Step 2.5.1.3

Add $1$ and $2$ .

$- 147 - 1 \cdot 2 h^{3} + h^{2}$

Step 2.5.2

Multiply $- 1$ by $2$ .

$- 147 - 2 h^{3} + h^{2}$