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Let's address each question in detail.

### Question 9

We need to determine which matrix multiplications are valid based on the dimensions provided:

- \( A \) is a \( 3 \times 2 \) matrix.
- \( B \) is a \( 2 \times 3 \) matrix.
- \( C \) is a \( 2 \times 2 \) matrix.

Matrix multiplication is only defined when the number of columns in the first matrix equals the number of rows in the second matrix.

Let's analyze each option:

1. **\( AB \)**: \( A \) is \( 3 \times 2 \) and \( B \) is \( 2 \times 3 \). The inner dimensions match (2), so \( AB \) is defined. Resulting matrix: \( 3 \times 3 \).
2. **\( BA \)**: \( B \) is \( 2 \times 3 \) and \( A \) is \( 3 \times 2 \). The inner dimensions match (3), so \( BA \) is defined. Resulting matrix: \( 2 \times 2 \).
3. **\( AC \)**: \( A \) is \( 3 \times 2 \) and \( C \) is \( 2 \times 2 \). The inner dimensions match (2), so \( AC \) is defined. Resulting matrix: \( 3 \times 2 \).
4. **\( CA \)**: \( C \) is \( 2 \times 2 \) and \( A \) is \( 3 \times 2 \). The inner dimensions do not match (2 vs. 3), so \( CA \) is *not defined*.
5. **\( BC \)**: \( B \) is \( 2 \times 3 \) and \( C \) is \( 2 \times 2 \). The inner dimensions do not match (3 vs. 2), so \( BC \) is *not defined*.
6. **\( CB \)**: \( C \) is \( 2 \times 2 \) and \( B \) is \( 2 \times 3 \). The inner dimensions match (2), so \( CB \) is defined. Resulting matrix: \( 2 \times 3 \).
7. **\( A^2 \)**: \( A \) is \( 3 \times 2 \). Squaring requires a square matrix, so \( A^2 \) is *not defined*.
8. **\( B^2 \)**: \( B \) is \( 2 \times 3 \). Squaring requires a square matrix, so \( B^2 \) is *not defined*.
9. **\( C^2 \)**: \( C \) is \( 2 \times 2 \), a square matrix. Squaring is possible, so \( C^2 \) is defined. Resulting matrix: \( 2 \times 2 \).

**Valid Multiplications**: \( AB \), \( BA \), \( AC \), \( CB \), \( C^2 \).

### Question 10

The given system of equations:

\[
\begin{cases}
-7x + 3y + 2z = -40 \\
6x - 7y + 3z = 43 \\
3x - 4y + z = 7
\end{cases}
\]

To write this system as an augmented matrix, we place the coefficients of \( x \), \( y \), and \( z \) in the first three columns, and the constants on the right side of each equation in the last column:

\[
\begin{bmatrix}
-7 & 3 & 2 & | & -40 \\
6 & -7 & 3 & | & 43 \\
3 & -4 & 1 & | & 7
\end{bmatrix}
\]

### Summary

1. **Valid Matrix Multiplications**: \( AB \), \( BA \), \( AC \), \( CB \), \( C^2 \).
2. **Augmented Matrix**:
   \[
   \begin{bmatrix}
   -7 & 3 & 2 & -40 \\
   6 & -7 & 3 & 43 \\
   3 & -4 & 1 & 7
   \end{bmatrix}
   \]

Would you like a further breakdown of any of these solutions or more examples?

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### Related Questions:

1. How do you determine if matrix multiplication is possible for matrices of arbitrary dimensions?
2. What is the purpose of an augmented matrix in solving systems of linear equations?
3. Why must matrices be square to compute powers like \( A^2 \)?
4. How does matrix multiplication affect the dimensions of the resulting matrix?
5. What are some common applications of augmented matrices in linear algebra?

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**Tip:** Always check the inner dimensions when determining the possibility of matrix multiplication; this simple step avoids common errors.

Question 9: Let A be a 3x2 matrix, B be a 2x3 matrix, and C be a 2x2 matrix. Which of the following matrix multiplications can be performed?

Question 10: Write the following system of equations as an augmented matrix:

-7x + 3y + 2z = -40
6x - 7y + 3z = 43
3x - 4y + z = 7

Determine which matrix multiplications are valid based on matrix dimensions for matrices A (3x2), B (2x3), and C (2x2). Learn to represent a system of linear equations as an augmented matrix. Suitable for students in Grades 10-12.

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