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Let's solve the matrix multiplications \( AB \) and \( BA \), where:

\[
A = \begin{bmatrix} -5 & 4 & 5 \\ 3 & 3 & -3 \end{bmatrix}
\]
and
\[
B = \begin{bmatrix} -3 & -4 \\ 0 & 1 \\ 0 & 3 \end{bmatrix}.
\]

### Step 1: Compute \( AB \)
To multiply \( A \) by \( B \), we perform the dot product of rows of \( A \) with columns of \( B \).

\[
AB = \begin{bmatrix} 
(-5)(-3) + (4)(0) + (5)(0) & (-5)(-4) + (4)(1) + (5)(3) \\ 
(3)(-3) + (3)(0) + (-3)(0) & (3)(-4) + (3)(1) + (-3)(3) 
\end{bmatrix}
\]

Simplifying the entries:

1. First row, first column: 
   \[
   (-5)(-3) + (4)(0) + (5)(0) = 15
   \]
2. First row, second column: 
   \[
   (-5)(-4) + (4)(1) + (5)(3) = 20 + 4 + 15 = 39
   \]
3. Second row, first column: 
   \[
   (3)(-3) + (3)(0) + (-3)(0) = -9
   \]
4. Second row, second column: 
   \[
   (3)(-4) + (3)(1) + (-3)(3) = -12 + 3 - 9 = -18
   \]

Thus, the result of \( AB \) is:
\[
AB = \begin{bmatrix} 15 & 39 \\ -9 & -18 \end{bmatrix}
\]

### Step 2: Compute \( BA \)
Now, multiply \( B \) by \( A \). The dimensions allow this operation since \( B \) is \( 3 \times 2 \) and \( A \) is \( 2 \times 3 \). The resulting matrix will be \( 3 \times 3 \).

\[
BA = \begin{bmatrix} 
(-3)(-5) + (-4)(3) & (-3)(4) + (-4)(3) & (-3)(5) + (-4)(-3) \\
(0)(-5) + (1)(3) & (0)(4) + (1)(3) & (0)(5) + (1)(-3) \\
(0)(-5) + (3)(3) & (0)(4) + (3)(3) & (0)(5) + (3)(-3)
\end{bmatrix}
\]

Simplifying the entries:

1. First row, first column: 
   \[
   (-3)(-5) + (-4)(3) = 15 - 12 = 3
   \]
2. First row, second column: 
   \[
   (-3)(4) + (-4)(3) = -12 - 12 = -24
   \]
3. First row, third column: 
   \[
   (-3)(5) + (-4)(-3) = -15 + 12 = -3
   \]
4. Second row, first column: 
   \[
   (0)(-5) + (1)(3) = 3
   \]
5. Second row, second column: 
   \[
   (0)(4) + (1)(3) = 3
   \]
6. Second row, third column: 
   \[
   (0)(5) + (1)(-3) = -3
   \]
7. Third row, first column: 
   \[
   (0)(-5) + (3)(3) = 9
   \]
8. Third row, second column: 
   \[
   (0)(4) + (3)(3) = 9
   \]
9. Third row, third column: 
   \[
   (0)(5) + (3)(-3) = -9
   \]

Thus, the result of \( BA \) is:
\[
BA = \begin{bmatrix} 3 & -24 & -3 \\ 3 & 3 & -3 \\ 9 & 9 & -9 \end{bmatrix}
\]

### Final Answer:
\[
AB = \begin{bmatrix} 15 & 39 \\ -9 & -18 \end{bmatrix}
\]
\[
BA = \begin{bmatrix} 3 & -24 & -3 \\ 3 & 3 & -3 \\ 9 & 9 & -9 \end{bmatrix}
\]

Would you like further details or clarification? Here are five related questions for practice:

1. What are the conditions for two matrices to be multiplied?
2. Can matrix multiplication be commutative? Why or why not?
3. How do the dimensions of the resulting matrix depend on the dimensions of the matrices being multiplied?
4. What happens if the number of columns in matrix \( A \) is not equal to the number of rows in matrix \( B \)?
5. What is the significance of the zero elements in matrices during multiplication?

**Tip:** Matrix multiplication is **not** commutative, meaning \( AB \neq BA \) in general. Always check dimensions and order carefully!

Perform matrix multiplication for the given matrices A and B: A = [-5 4 5; 3 3 -3] and B = [-3 -4; 0 1; 0 3].

This problem demonstrates how to perform matrix multiplication between two matrices, A (2x3) and B (3x2). Step-by-step calculations are shown, providing the product matrices AB and BA.

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