https://file.aiquickdraw.com/mathbot/file/1727720079189v1p00c0a.jpg

Let's analyze the matrix operations from your image. The matrices are:

\[
A = \begin{bmatrix} 2 & -1 \\ 0 & 6 \\ 9 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -3 & 1 \\ 8 & 0 & -7 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & -2 & 5 \\ 2 & 1 & 0 \\ 4 & -3 & 1 \end{bmatrix}
\]

\[
D = \begin{bmatrix} 5 & -3 \\ 1 & 13 \\ 7 & -1 \end{bmatrix}, \quad E = \begin{bmatrix} 11 \\ 11 \\ -4 \end{bmatrix}
\]

### Matrix Dimensions:
- \( A \) is \( 3 \times 2 \)
- \( B \) is \( 2 \times 3 \)
- \( C \) is \( 3 \times 3 \)
- \( D \) is \( 3 \times 2 \)
- \( E \) is \( 3 \times 1 \)

### Operations:
1. **CA**: To multiply \( C \times A \), the number of columns in \( C \) must match the number of rows in \( A \). Both are \( 3 \times 3 \) and \( 3 \times 2 \), so this multiplication is **not possible**.
   
2. **DC**: \( D \) is \( 3 \times 2 \) and \( C \) is \( 3 \times 3 \). Since the number of columns in \( D \) (2) does not match the number of rows in \( C \) (3), this is **not possible**.

3. **C + A**: To add two matrices, they must have the same dimensions. \( C \) is \( 3 \times 3 \), and \( A \) is \( 3 \times 2 \), so this addition is **not possible**.

4. **CE**: \( C \) is \( 3 \times 3 \), and \( E \) is \( 3 \times 1 \). The number of columns in \( C \) matches the number of rows in \( E \), so this multiplication is **possible**.

5. **BE**: \( B \) is \( 2 \times 3 \), and \( E \) is \( 3 \times 1 \). The number of columns in \( B \) matches the number of rows in \( E \), so this multiplication is **possible**.

6. **AB**: \( A \) is \( 3 \times 2 \), and \( B \) is \( 2 \times 3 \). The number of columns in \( A \) matches the number of rows in \( B \), so this multiplication is **possible**.

7. **EC**: \( E \) is \( 3 \times 1 \), and \( C \) is \( 3 \times 3 \). The number of columns in \( E \) (1) does not match the number of rows in \( C \) (3), so this multiplication is **not possible**.

8. **D - A**: To subtract two matrices, they must have the same dimensions. \( D \) and \( A \) are both \( 3 \times 2 \), so this subtraction is **possible**.

### Summary of Meaningful Operations:
- \( CE \)
- \( BE \)
- \( AB \)
- \( D - A \)

Would you like more details or further explanation on any of the operations?

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Here are some related questions to deepen understanding:
1. What are the general rules for matrix multiplication?
2. How do you determine if two matrices can be added or subtracted?
3. Can you explain the dimensions of matrices involved in real-world applications?
4. How does matrix transposition affect possible operations?
5. What are some common errors made when performing matrix operations?

**Tip**: Always check the dimensions of matrices before attempting to perform any operations like addition, subtraction, or multiplication.

Which of the following matrix operations are meaningful? Given matrices A (3x2), B (2x3), C (3x3), D (3x2), and E (3x1).

Explore meaningful matrix operations with matrices A (3x2), B (2x3), C (3x3), D (3x2), and E (3x1). Learn about matrix multiplication and addition rules. Suitable for high school students in Grades 10-12.

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