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Thanks for sharing the worksheet! Let's go through each problem in parts.

### Problem 1: Verify if \( AB \neq BA \) for the given matrices.

For each pair of matrices \( A \) and \( B \), we need to compute both \( AB \) and \( BA \), then check if they are equal.

- (a) \( A = \begin{bmatrix} -1 & 2 & -4 \\ 3 & -4 & 3 \\ 1 & -2 & 5 \end{bmatrix} \), \( B = \begin{bmatrix} 3 & -2 & 1 \\ -2 & 1 & 2 \\ 4 & -5 & 4 \end{bmatrix} \)
- (b) \( A = \begin{bmatrix} 6 & -1 & 3 \\ -1 & 5 & -3 \\ 4 & 1 & 6 \end{bmatrix} \), \( B = \begin{bmatrix} -5 & 4 & 0 \\ 6 & -2 & -5 \\ 6 & 1 & -2 \end{bmatrix} \)
- (c) \( A = \begin{bmatrix} 1 & 3 & 4 & -1 \\ 2 & -4 & 3 & 2 \\ -1 & 2 & 5 & 0 \\ -2 & 1 & 3 & 6 \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 2 & 1 & -2 \\ 3 & -4 & -2 & 1 \\ 4 & 3 & 5 & 3 \\ -1 & 2 & 4 & 6 \end{bmatrix} \)
- (d) \( A = \begin{bmatrix} 0 & -2 & 3 & 1 \\ -1 & 7 & 1 & -2 \\ 2 & -3 & -4 & 6 \\ -3 & 3 & 4 & 8 \end{bmatrix} \), \( B = \begin{bmatrix} -2 & 0 & -1 & 2 \\ -5 & -3 & 4 & 0 \\ 1 & 4 & -4 & 1 \\ 5 & 2 & -3 & 2 \end{bmatrix} \)

### Problem 2: Calculate the product \( AB \).

This involves matrix multiplication for the provided matrices.

- (a) \( A = \begin{bmatrix} -2 & 3 \\ 5 & -2 \end{bmatrix} \), \( B = \begin{bmatrix} 4 & 2 \\ -5 & 3 \\ 6 & 7 \end{bmatrix} \)
- (b) \( A = \begin{bmatrix} 5 & 4 & 9 \end{bmatrix} \), \( B = \begin{bmatrix} 3 & -2 & 1 \\ -2 & 1 & 2 \\ 4 & -5 & 4 \end{bmatrix} \)
- (c) \( A = \begin{bmatrix} 4 & -1 \\ 1 & 3 \\ -6 & 7 \end{bmatrix} \), \( B = \begin{bmatrix} 8 & -4 & 5 \\ -7 & 2 & 1 \end{bmatrix} \)
- (d) \( A = \begin{bmatrix} -5 & 4 & 0 \\ 6 & -2 & 5 \\ 6 & 1 & -2 \end{bmatrix} \), \( B = \begin{bmatrix} 7 \\ 4 \\ 2 \end{bmatrix} \)

### Problem 3: Calculate the determinant of the given matrices.

We need to compute the determinant for each matrix.

- (a) \( A = \begin{bmatrix} 5 & 3 \\ -1 & 4 \end{bmatrix} \)
- (b) \( B = \begin{bmatrix} 6 & -1 & 3 \\ -1 & 5 & -3 \\ 4 & 1 & 6 \end{bmatrix} \)
- (c) \( C = \begin{bmatrix} -5 & 4 & 0 \\ 6 & -2 & 5 \\ 6 & 1 & -2 \end{bmatrix} \)

I can start working through specific calculations for any part you prefer. Which problem would you like me to solve first?

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Explore matrix operations including multiplication, checking matrix equality, and determinant calculation with examples from a practice worksheet. This guide covers mathematical concepts like matrix multiplication rules, determinant formulas, and their properties.

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