https://r2.erweima.ai/mathbot/image/1726420050284fdrnopo9.jpg

We are given matrices \( A \) and \( B \) and asked to find their product \( AB \), then verify which of the provided answers is correct.

Matrix \( A \) is:

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

Matrix \( B \) is:

\[
B = \begin{bmatrix}
6 & 7 \\
8 & 9
\end{bmatrix}
\]

### Step-by-step matrix multiplication:
To calculate \( AB \), we need to compute the product of each row of \( A \) with each column of \( B \). The result will be a \( 3 \times 2 \) matrix since \( A \) is \( 3 \times 2 \) and \( B \) is \( 2 \times 2 \).

\[
AB = \begin{bmatrix}
(0 \cdot 6 + 3 \cdot 8) & (0 \cdot 7 + 3 \cdot 9) \\
(1 \cdot 6 + 4 \cdot 8) & (1 \cdot 7 + 4 \cdot 9) \\
(2 \cdot 6 + 5 \cdot 8) & (2 \cdot 7 + 5 \cdot 9)
\end{bmatrix}
\]

#### First row of \( AB \):
\[
(0 \cdot 6 + 3 \cdot 8) = 0 + 24 = 24
\]
\[
(0 \cdot 7 + 3 \cdot 9) = 0 + 27 = 27
\]

#### Second row of \( AB \):
\[
(1 \cdot 6 + 4 \cdot 8) = 6 + 32 = 38
\]
\[
(1 \cdot 7 + 4 \cdot 9) = 7 + 36 = 43
\]

#### Third row of \( AB \):
\[
(2 \cdot 6 + 5 \cdot 8) = 12 + 40 = 52
\]
\[
(2 \cdot 7 + 5 \cdot 9) = 14 + 45 = 59
\]

So, the resulting matrix \( AB \) is:

\[
AB = \begin{bmatrix}
24 & 27 \\
38 & 43 \\
52 & 59
\end{bmatrix}
\]

### Correct Answer:
The correct matrix multiplication result matches option **E** in the image provided:

\[
A \times B = \begin{bmatrix}
24 & 27 \\
38 & 43 \\
52 & 59
\end{bmatrix}
\]

Would you like more details about matrix multiplication or have any further questions?

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Here are 5 related questions to expand your knowledge:
1. How is matrix multiplication different from element-wise multiplication?
2. What is the significance of the dimensions of matrices in multiplication?
3. What happens if we multiply two square matrices?
4. How can matrix multiplication be used in transformations, such as rotations in geometry?
5. Can non-square matrices have an inverse? If not, why?

**Tip:** Always check matrix dimensions before attempting multiplication; they must align (columns of the first matrix equal to rows of the second).

Given two matrices A = [[0, 3], [1, 4], [2, 5]] and B = [[6, 7], [8, 9]], find the product AB and determine the correct result from the options provided.

Learn how to multiply matrices A and B, where A is a 3x2 matrix and B is a 2x2 matrix. This detailed guide covers the steps to calculate the product and verify the correct solution among multiple choices.

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