Linear Algebra Examples

$[\begin{array}{c} 2 \cdot k - m \cdot x^{2} & - k \\ - k & k - \frac{m}{2} \cdot x^{2} \end{array}]$

Step 1

Multiply $- m$ by $x^{2}$ .

$[\begin{array}{c} 2 k - m x^{2} & - k \\ - k & k - \frac{m}{2} \cdot x^{2} \end{array}]$

Step 2

Combine $x^{2}$ and $\frac{m}{2}$ .

$[\begin{array}{c} 2 k - m x^{2} & - k \\ - k & k - \frac{x^{2} m}{2} \end{array}]$

Step 3

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

$(2 k - m x^{2}) (k - \frac{x^{2} m}{2}) - (- k (- k))$

Step 4

Simplify the determinant.

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Step 4.1

Simplify each term.

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Step 4.1.1

Expand $(2 k - m x^{2}) (k - \frac{x^{2} m}{2})$ using the FOIL Method.

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Step 4.1.1.1

Apply the distributive property.

$2 k (k - \frac{x^{2} m}{2}) - m x^{2} (k - \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.1.2

Apply the distributive property.

$2 k \cdot k + 2 k (- \frac{x^{2} m}{2}) - m x^{2} (k - \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.1.3

Apply the distributive property.

$2 k \cdot k + 2 k (- \frac{x^{2} m}{2}) - m x^{2} k - m x^{2} (- \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.2

Simplify and combine like terms.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Multiply $k$ by $k$ by adding the exponents.

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Step 4.1.2.1.1.1

Move $k$ .

$2 (k \cdot k) + 2 k (- \frac{x^{2} m}{2}) - m x^{2} k - m x^{2} (- \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.2.1.1.2

Multiply $k$ by $k$ .

$2 k^{2} + 2 k (- \frac{x^{2} m}{2}) - m x^{2} k - m x^{2} (- \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.2.1.2

Cancel the common factor of $2$ .

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Step 4.1.2.1.2.1

Move the leading negative in $- \frac{x^{2} m}{2}$ into the numerator.

$2 k^{2} + 2 k \frac{- x^{2} m}{2} - m x^{2} k - m x^{2} (- \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.2.1.2.2

Factor $2$ out of $2 k$ .

$2 k^{2} + 2 (k) \frac{- x^{2} m}{2} - m x^{2} k - m x^{2} (- \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.2.1.2.3

Cancel the common factor.

$2 k^{2} + 2 k \frac{- x^{2} m}{2} - m x^{2} k - m x^{2} (- \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.2.1.2.4

Rewrite the expression.

$2 k^{2} + k (- x^{2} m) - m x^{2} k - m x^{2} (- \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.2.1.3

Rewrite using the commutative property of multiplication.

$2 k^{2} - k x^{2} m - m x^{2} k - m x^{2} (- \frac{x^{2} m}{2}) - (- k (- k))$

Step 4.1.2.1.4

Multiply $- m x^{2} (- \frac{x^{2} m}{2})$ .

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Step 4.1.2.1.4.1

Multiply $- 1$ by $- 1$ .

$2 k^{2} - k x^{2} m - m x^{2} k + 1 m x^{2} \frac{x^{2} m}{2} - (- k (- k))$

Step 4.1.2.1.4.2

Multiply $m$ by $1$ .

$2 k^{2} - k x^{2} m - m x^{2} k + m x^{2} \frac{x^{2} m}{2} - (- k (- k))$

Step 4.1.2.1.4.3

Combine $m$ and $\frac{x^{2} m}{2}$ .

$2 k^{2} - k x^{2} m - m x^{2} k + x^{2} \frac{m (x^{2} m)}{2} - (- k (- k))$

Step 4.1.2.1.4.4

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

$2 k^{2} - k x^{2} m - m x^{2} k + x^{2} \frac{m^{1} m x^{2}}{2} - (- k (- k))$

Step 4.1.2.1.4.5

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

$2 k^{2} - k x^{2} m - m x^{2} k + x^{2} \frac{m^{1} m^{1} x^{2}}{2} - (- k (- k))$

Step 4.1.2.1.4.6

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

$2 k^{2} - k x^{2} m - m x^{2} k + x^{2} \frac{m^{1 + 1} x^{2}}{2} - (- k (- k))$

Step 4.1.2.1.4.7

Add $1$ and $1$ .

$2 k^{2} - k x^{2} m - m x^{2} k + x^{2} \frac{m^{2} x^{2}}{2} - (- k (- k))$

Step 4.1.2.1.4.8

Combine $x^{2}$ and $\frac{m^{2} x^{2}}{2}$ .

$2 k^{2} - k x^{2} m - m x^{2} k + \frac{x^{2} (m^{2} x^{2})}{2} - (- k (- k))$

Step 4.1.2.1.4.9

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

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Step 4.1.2.1.4.9.1

Move $x^{2}$ .

$2 k^{2} - k x^{2} m - m x^{2} k + \frac{x^{2} x^{2} m^{2}}{2} - (- k (- k))$

Step 4.1.2.1.4.9.2

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

$2 k^{2} - k x^{2} m - m x^{2} k + \frac{x^{2 + 2} m^{2}}{2} - (- k (- k))$

Step 4.1.2.1.4.9.3

Add $2$ and $2$ .

$2 k^{2} - k x^{2} m - m x^{2} k + \frac{x^{4} m^{2}}{2} - (- k (- k))$

Step 4.1.2.2

Move $m$ .

$2 k^{2} - k m x^{2} - 1 k m x^{2} + \frac{x^{4} m^{2}}{2} - (- k (- k))$

Step 4.1.2.3

Subtract $k m x^{2}$ from $- k m x^{2}$ .

$2 k^{2} - 2 k m x^{2} + \frac{x^{4} m^{2}}{2} - (- k (- k))$

Step 4.1.3

Multiply $k$ by $k$ by adding the exponents.

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Step 4.1.3.1

Move $k$ .

$2 k^{2} - 2 k m x^{2} + \frac{x^{4} m^{2}}{2} - (- (k \cdot k) \cdot - 1)$

Step 4.1.3.2

Multiply $k$ by $k$ .

$2 k^{2} - 2 k m x^{2} + \frac{x^{4} m^{2}}{2} - (- k^{2} \cdot - 1)$

Step 4.1.4

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

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Step 4.1.4.1

Multiply $- 1$ by $- 1$ .

$2 k^{2} - 2 k m x^{2} + \frac{x^{4} m^{2}}{2} - (1 k^{2})$

Step 4.1.4.2

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

$2 k^{2} - 2 k m x^{2} + \frac{x^{4} m^{2}}{2} - k^{2}$

Step 4.2

Subtract $k^{2}$ from $2 k^{2}$ .

$k^{2} - 2 k m x^{2} + \frac{x^{4} m^{2}}{2}$