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In the image, you are tasked with finding the matrix products of two matrices \( A \) and \( B \). The matrices are given as follows:

\[
A = \begin{pmatrix} 
-2 & 2 & 5 \\
5 & 5 & 3 \\
-2 & 7 & 2
\end{pmatrix}
\quad \text{and} \quad
B = \begin{pmatrix} 
-4 & 2 \\
9 & 9 \\
-3 & 2
\end{pmatrix}
\]

The problem requires calculating:
- (a) \( A \times B \)
- (b) \( B \times A \)

### (a) Calculating \( A \times B \)
Matrix multiplication is defined as follows: for matrices \( A \) (size \( m \times n \)) and \( B \) (size \( n \times p \)), the product matrix \( C \) (size \( m \times p \)) has elements \( C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} \).

Here, \( A \) is a \( 3 \times 3 \) matrix and \( B \) is a \( 3 \times 2 \) matrix. The result of \( A \times B \) will be a \( 3 \times 2 \) matrix.

Now, let's calculate each element of the resulting matrix:

\[
A \times B = \begin{pmatrix} 
(-2)(-4) + (2)(9) + (5)(-3) & (-2)(2) + (2)(9) + (5)(2) \\
(5)(-4) + (5)(9) + (3)(-3) & (5)(2) + (5)(9) + (3)(2) \\
(-2)(-4) + (7)(9) + (2)(-3) & (-2)(2) + (7)(9) + (2)(2)
\end{pmatrix}
\]

Calculating the elements:

- First row:
  - \( (1,1) = 8 + 18 - 15 = 11 \)
  - \( (1,2) = -4 + 18 + 10 = 24 \)
- Second row:
  - \( (2,1) = -20 + 45 - 9 = 16 \)
  - \( (2,2) = 10 + 45 + 6 = 61 \)
- Third row:
  - \( (3,1) = 8 + 63 - 6 = 65 \)
  - \( (3,2) = -4 + 63 + 4 = 63 \)

Thus, 
\[
A \times B = \begin{pmatrix} 
11 & 24 \\
16 & 61 \\
65 & 63 
\end{pmatrix}
\]

### (b) Calculating \( B \times A \)
Now, matrix \( B \times A \) is the multiplication of a \( 3 \times 2 \) matrix with a \( 3 \times 3 \) matrix. However, this multiplication is **not defined** because the number of columns in \( B \) (which is 2) does not match the number of rows in \( A \) (which is 3).

Thus:
- \( B \times A \) is not possible.

Would you like any further details or explanations on this?

### Additional Questions:
1. How does the matrix multiplication rule work in terms of dimensions?
2. Can you explain when matrix multiplication is undefined?
3. What are some special properties of matrix products?
4. How can matrix multiplication be applied in real-world problems?
5. What happens when we multiply square matrices?

### Tip:
When multiplying two matrices, always check if the number of columns in the first matrix matches the number of rows in the second matrix to determine if the multiplication is possible.

Determine the matrix products for the given matrices A and B:
A = [[-2, 2, 5], [5, 5, 3], [-2, 7, 2]]
B = [[-4, 2], [9, 9], [-3, 2]]

Tasks:
a. A × B
b. B × A

Learn how to multiply two matrices, A and B, with detailed step-by-step solutions. Matrix A is 3x3, and matrix B is 3x2. The tasks include finding A × B and determining why B × A is undefined due to dimensional incompatibility. Suitable for high school students studying linear algebra.

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