https://file.aiquickdraw.com/mathbot/file/1726711123168dwt63ttp.jpg

Let's go through the two problems in the image:

### Problem 2:
You are given the matrix equation:
\[
A - Y = B
\]
where:
\[
A = \begin{pmatrix} 5 & -2 \\ -3 & 1 \end{pmatrix}, \quad Y = ? , \quad B = \begin{pmatrix} 2 & -3 \\ 0 & -1 \end{pmatrix}
\]
You are asked to find matrix \( Y \).

To find \( Y \), rearrange the equation:
\[
Y = A - B
\]
Now, subtract matrix \( B \) from matrix \( A \):
\[
Y = \begin{pmatrix} 5 & -2 \\ -3 & 1 \end{pmatrix} - \begin{pmatrix} 2 & -3 \\ 0 & -1 \end{pmatrix}
\]
Perform the subtraction element-wise:
\[
Y = \begin{pmatrix} 5 - 2 & -2 - (-3) \\ -3 - 0 & 1 - (-1) \end{pmatrix}
\]
\[
Y = \begin{pmatrix} 3 & 1 \\ -3 & 2 \end{pmatrix}
\]

### Problem 3:
You are asked to find the product of two matrices:
\[
\begin{pmatrix} 2 & -1 \\ 1 & 3 \end{pmatrix} \times \begin{pmatrix} 2 \\ 4 \end{pmatrix}
\]
This is a matrix multiplication of a \( 2 \times 2 \) matrix with a \( 2 \times 1 \) column vector. Perform the multiplication as follows:

1. Multiply the first row of the matrix by the column vector:
   \[
   (2 \times 2) + (-1 \times 4) = 4 - 4 = 0
   \]

2. Multiply the second row of the matrix by the column vector:
   \[
   (1 \times 2) + (3 \times 4) = 2 + 12 = 14
   \]

Thus, the product is:
\[
\begin{pmatrix} 0 \\ 14 \end{pmatrix}
\]

### Summary of Results:
- **Problem 2**: \( Y = \begin{pmatrix} 3 & 1 \\ -3 & 2 \end{pmatrix} \)
- **Problem 3**: The product is \( \begin{pmatrix} 0 \\ 14 \end{pmatrix} \).

Would you like further details or have any additional questions?

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Here are 5 related questions for further exploration:
1. What is the general rule for adding or subtracting matrices?
2. How is matrix multiplication performed when matrices are not square?
3. What are the properties of matrix inverses and when can they be used?
4. How do you determine if two matrices are conformable for multiplication?
5. Can you extend matrix operations to solve systems of linear equations?

**Tip**: When performing matrix operations, always ensure the dimensions are compatible to avoid errors.

Find matrix Y and perform the matrix multiplication shown in the image.

Learn how to solve matrix problems involving matrix subtraction (Y = A - B) and matrix multiplication in this detailed step-by-step solution. This problem covers key matrix arithmetic concepts and demonstrates both operations with clear explanations. Suitable for high school students in Grades 10-12.

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