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: