https://file.aiquickdraw.com/mathbot/file/1729849123278l7xgsk0w.jpg

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.

Find the determinants of the matrices A = [[1, 1], [4, -2]] and B = [[4, -8], [1, -4]].

This problem explores the calculation of determinants for matrices A and B. Learn step-by-step how to find the determinant of a 2x2 matrix using the formula det(A) = ad - bc. Suitable for high school students studying linear algebra.

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