https://file.aiquickdraw.com/mathbot/file/1725536262121mxlreek2.jpg

The question asks to find \( XY \), where:

\[
X = \begin{bmatrix} \frac{2}{4} & \frac{-3}{6} \end{bmatrix} 
\quad \text{and} \quad 
Y = \begin{bmatrix} \frac{9}{6} \\ \frac{-6}{4} \end{bmatrix}
\]

We are tasked to compute the matrix product \( XY \).

### Step-by-Step Solution:
1. First, simplify the elements of the matrices:
   \[
   X = \begin{bmatrix} \frac{2}{4} & \frac{-3}{6} \end{bmatrix} = \begin{bmatrix} 0.5 & -0.5 \end{bmatrix}
   \quad \text{and} \quad 
   Y = \begin{bmatrix} \frac{9}{6} \\ \frac{-6}{4} \end{bmatrix} = \begin{bmatrix} 1.5 \\ -1.5 \end{bmatrix}
   \]

2. Now, multiply the row vector \( X \) by the column vector \( Y \):
   \[
   XY = (0.5)(1.5) + (-0.5)(-1.5)
   \]

3. Compute each term:
   \[
   (0.5)(1.5) = 0.75 \quad \text{and} \quad (-0.5)(-1.5) = 0.75
   \]

4. Add the results:
   \[
   XY = 0.75 + 0.75 = 1.5
   \]

Thus, the product \( XY = 1.5 \).

Would you like further details or have any questions?

Here are five related questions:
1. How do you multiply two matrices with different dimensions?
2. Can you find the inverse of matrix \( X \) if it were a square matrix?
3. How does matrix multiplication differ from regular multiplication?
4. What properties must matrices have for multiplication to be defined?
5. How would this process change if \( X \) and \( Y \) were 3x3 matrices?

**Tip:** When multiplying matrices, ensure that the number of columns in the first matrix matches the number of rows in the second matrix to get a valid product.

Given that X = [2/4 -3/6] and Y = [9/6 -6/4], find XY.

Learn how to find the product of matrices X = [2/4 -3/6] and Y = [9/6 -6/4]. This detailed solution covers matrix multiplication, fraction simplification, and relevant theorems. Suitable for students in Grades 9-12.

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