The problem asks to find the determinants of matrices $A$ and $B$, where:

Matrix $A$ is: $A = \begin{bmatrix} 1 & 1 \\ 4 & -2 \end{bmatrix}$

Matrix $B$ is: $B = \begin{bmatrix} 4 & -8 \\ 1 & -4 \end{bmatrix}$

Determinant of $A$:

The determinant of a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is calculated as: $\text{det}(A) = ad - bc$ For matrix $A$: $\text{det}(A) = (1)(-2) - (1)(4) = -2 - 4 = -6$

Determinant of $B$:

Similarly, for matrix $B$: $\text{det}(B) = (4)(-4) - (-8)(1) = -16 + 8 = -8$

Final Results:

$\text{det}(A) = -6, \quad \text{det}(B) = -8$

Let me know if you want further details on how determinants work, or if you have any questions!

Related Questions:

1. How do we calculate the determinant of a 3x3 matrix?
2. What are the properties of determinants?
3. How can we use the determinant to determine if a matrix is invertible?
4. What is the effect of row operations on the determinant of a matrix?
5. How does changing an element of a matrix affect its determinant?

Tip: The determinant of a matrix can indicate if the matrix has an inverse. A non-zero determinant means the matrix is invertible.