https://file.aiquickdraw.com/mathbot/file/1728274590096h4ltchrb.jpg

The image you've uploaded contains two matrices, \( A \) and \( B \), and asks for the product of:

a) \( A \times B \)

b) \( B \times A \)

Let's begin by defining the matrices:

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

### Step-by-step process for \( A \times B \):

Matrix multiplication is defined as:

\[
(A \times B)_{ij} = \sum_k A_{ik} B_{kj}
\]

Where \( i \) and \( j \) are the row and column indices, respectively.

1. **Dimensions of \( A \)**: \( 3 \times 3 \)
2. **Dimensions of \( B \)**: \( 3 \times 2 \)

The resulting matrix \( A \times B \) will have dimensions \( 3 \times 2 \).

Let's compute \( A \times B \):

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

Simplifying each element:

\[
A \times B = \begin{pmatrix}
8 + 18 - 15 & -4 + 18 + 45 \\
-20 + 45 - 9 & 10 + 45 + 27 \\
8 + 63 - 6 & -4 + 63 + 18
\end{pmatrix}
\]

This gives:

\[
A \times B = \begin{pmatrix}
11 & 59 \\
16 & 82 \\
65 & 77
\end{pmatrix}
\]

### Step-by-step process for \( B \times A \):

Next, we'll compute \( B \times A \). Since:

- \( B \) is \( 3 \times 2 \)
- \( A \) is \( 3 \times 3 \)

Multiplication \( B \times A \) is **not possible**, because the number of columns in \( B \) (which is 2) does not match the number of rows in \( A \) (which is 3). Therefore, **\( B \times A \) is undefined**.

Would you like more details or clarification on any step?

### Related Questions:
1. What is the general rule for matrix multiplication?
2. How do you verify if two matrices can be multiplied?
3. What are the properties of matrix multiplication, such as commutativity?
4. How can you find the inverse of a matrix, and when does it exist?
5. What is the determinant of a matrix, and how is it useful?

### Tip:
When multiplying matrices, always check the dimensions of the matrices first. For two matrices \( A \times B \), the number of columns in \( A \) must match the number of rows in \( B \).

Matrix multiplication problem where the user is asked to compute A × B and B × A for two matrices A and B.

This problem explores matrix multiplication, where students are asked to compute A × B and determine if B × A is possible. Matrix A is a 3×3 matrix, and matrix B is a 3×2 matrix. Learn the step-by-step process of solving A × B and why B × A is undefined due to dimension mismatch. Suitable for high school students.

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