https://file.aiquickdraw.com/mathbot/file/1725536610511j9a9omiv.jpg

The image contains two matrices, \( A \) and \( B \), and a task asking to find the products \( AB \) and \( BA \). The matrices are given as:

\[
A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 1 \\ 0 & 3 \end{bmatrix}
\]

### Step 1: Matrix Multiplication \( AB \)

Matrix multiplication follows the rule that the element at position \( (i, j) \) in the product is the dot product of the \( i \)-th row of the first matrix with the \( j \)-th column of the second matrix. Let's calculate \( AB \).

\[
AB = \begin{bmatrix}
2 \cdot 5 + 3 \cdot 0 & 2 \cdot 1 + 3 \cdot 3 \\
1 \cdot 5 + 4 \cdot 0 & 1 \cdot 1 + 4 \cdot 3
\end{bmatrix}
= \begin{bmatrix}
10 & 11 \\
5 & 13
\end{bmatrix}
\]

### Step 2: Matrix Multiplication \( BA \)

Now, let's calculate \( BA \).

\[
BA = \begin{bmatrix}
5 \cdot 2 + 1 \cdot 1 & 5 \cdot 3 + 1 \cdot 4 \\
0 \cdot 2 + 3 \cdot 1 & 0 \cdot 3 + 3 \cdot 4
\end{bmatrix}
= \begin{bmatrix}
11 & 19 \\
3 & 12
\end{bmatrix}
\]

### Final Results:
\[
AB = \begin{bmatrix} 10 & 11 \\ 5 & 13 \end{bmatrix}, \quad BA = \begin{bmatrix} 11 & 19 \\ 3 & 12 \end{bmatrix}
\]

Let me know if you'd like further details or have any questions.

Here are 5 related questions to expand on this:

1. How do you determine when matrix multiplication is possible?
2. Can matrix multiplication be commutative? Why or why not?
3. What are the properties of matrix multiplication in terms of associativity and distributivity?
4. How can we compute the determinant of the matrices \( A \) and \( B \)?
5. What happens when you multiply a matrix by the identity matrix?

### Tip: 
Matrix multiplication is not always commutative, meaning \( AB \neq BA \), as shown in this example. Always check the dimensions carefully before multiplying!

Find AB and BA for matrices A = [[2, 3], [1, 4]] and B = [[5, 1], [0, 3]].

Learn how to multiply two 2x2 matrices, A and B. This problem involves matrix multiplication and shows how AB is different from BA. Suitable for Grades 9-12 or early university linear algebra courses.

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