https://r2.erweima.ai/mathbot/image/1726126714693nmmm7anj.jpg

In the image, there are two matrix-based exercises.

### Problem 1:
Given the matrices:
- \( A = \begin{bmatrix} 7 & -1 \\ 6 & 9 \end{bmatrix} \)
- \( B = \begin{bmatrix} 0 & 4 \\ 3 & -2 \end{bmatrix} \)
- \( C = \begin{bmatrix} 8 & 3 \\ 6 & 1 \end{bmatrix} \)

You are asked to find the following:
1. \( A + B \)
2. \( C - A \)
3. \( 3A \)
4. \( 4B + 2C \)

### Problem 2:
Given the matrices:
- \( A = \begin{bmatrix} 2 & 8 \\ 3 & 0 \\ 5 & 1 \end{bmatrix} \)
- \( B = \begin{bmatrix} 2 & 0 \\ 3 & 8 \end{bmatrix} \)
- \( C = \begin{bmatrix} 7 & 2 \\ 6 & 3 \end{bmatrix} \)

No specific operations are mentioned in Problem 2, but it seems like operations involving these matrices could follow a similar pattern to Problem 1, such as matrix addition, subtraction, or scalar multiplication.

I'll begin solving the operations from **Problem 1**:

#### 1(a): \( A + B \)
\[
A + B = \begin{bmatrix} 7 & -1 \\ 6 & 9 \end{bmatrix} + \begin{bmatrix} 0 & 4 \\ 3 & -2 \end{bmatrix} = \begin{bmatrix} 7 + 0 & -1 + 4 \\ 6 + 3 & 9 + (-2) \end{bmatrix} = \begin{bmatrix} 7 & 3 \\ 9 & 7 \end{bmatrix}
\]

#### 1(b): \( C - A \)
\[
C - A = \begin{bmatrix} 8 & 3 \\ 6 & 1 \end{bmatrix} - \begin{bmatrix} 7 & -1 \\ 6 & 9 \end{bmatrix} = \begin{bmatrix} 8 - 7 & 3 - (-1) \\ 6 - 6 & 1 - 9 \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ 0 & -8 \end{bmatrix}
\]

#### 1(c): \( 3A \) (Scalar multiplication)
\[
3A = 3 \times \begin{bmatrix} 7 & -1 \\ 6 & 9 \end{bmatrix} = \begin{bmatrix} 3 \times 7 & 3 \times (-1) \\ 3 \times 6 & 3 \times 9 \end{bmatrix} = \begin{bmatrix} 21 & -3 \\ 18 & 27 \end{bmatrix}
\]

#### 1(d): \( 4B + 2C \)
\[
4B = 4 \times \begin{bmatrix} 0 & 4 \\ 3 & -2 \end{bmatrix} = \begin{bmatrix} 0 & 16 \\ 12 & -8 \end{bmatrix}
\]
\[
2C = 2 \times \begin{bmatrix} 8 & 3 \\ 6 & 1 \end{bmatrix} = \begin{bmatrix} 16 & 6 \\ 12 & 2 \end{bmatrix}
\]
\[
4B + 2C = \begin{bmatrix} 0 & 16 \\ 12 & -8 \end{bmatrix} + \begin{bmatrix} 16 & 6 \\ 12 & 2 \end{bmatrix} = \begin{bmatrix} 0 + 16 & 16 + 6 \\ 12 + 12 & -8 + 2 \end{bmatrix} = \begin{bmatrix} 16 & 22 \\ 24 & -6 \end{bmatrix}
\]

Would you like me to solve Problem 2 or clarify any part of this solution?

Here are 5 related questions for deeper understanding:
1. How do you determine if two matrices can be added or subtracted?
2. What is the geometric interpretation of multiplying a matrix by a scalar?
3. How does matrix addition differ from element-wise operations in matrices?
4. What are the properties of scalar multiplication in matrix algebra?
5. How does matrix subtraction impact systems of linear equations?

**Tip**: Always check the dimensions of matrices before attempting operations like addition, subtraction, or multiplication, as incompatible dimensions can make the operation impossible.

Matrix operations including addition, subtraction, and scalar multiplication with matrices A, B, and C.

This page provides step-by-step solutions for matrix operations, including the addition, subtraction, and scalar multiplication of given matrices A, B, and C. Learn about key matrix algebra concepts and explore how these operations are performed.

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