To find the determinant of the $2 \times 2$ matrix:

$\begin{bmatrix} 6 & 9 \\ 4 & 2 \end{bmatrix}$

The formula for the determinant of a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is:

$\text{det} = ad - bc$

For this matrix:

• $a = 6$
• $b = 9$
• $c = 4$
• $d = 2$

Substitute these values into the formula:

$\text{det} = (6 \cdot 2) - (9 \cdot 4) = 12 - 36 = -24$

Therefore, the correct answer is -24.

Would you like further details or have any questions?

Here are five related questions to expand on this topic:

1. How do you find the determinant of a $3 \times 3$ matrix?
2. What does it mean if the determinant of a matrix is zero?
3. How is the determinant used in solving systems of linear equations?
4. Can determinants be calculated for matrices larger than $2 \times 2$?
5. What are the properties of determinants when matrices are multiplied?

Tip: For a quick determinant check of $2 \times 2$ matrices, remember the formula $ad - bc$ to avoid calculation errors.