Evaluate the determinant of the matrix $\begin{bmatrix} 5 & -3 \\ 2 & -4 \end{bmatrix}$.

Solution:

First, write the matrix in determinant form using vertical bars:

$\begin{vmatrix} 5 & -3 \\ 2 & -4 \end{vmatrix}$

Next, calculate the determinant by subtracting the product of the off-diagonal entries from the product of the main diagonal entries:

$(5)(-4) - (2)(-3)$

Now, simplify the expression by performing the multiplications:

$-20 - (-6)$

Finally, convert the subtraction to addition and solve:

$-20 + 6 = -14$

The determinant is $-14$.