https://r2.erweima.ai/mathbot/image/17268802711730xq175o9.jpg

Let's solve the matrix problems provided in the image.

### Actividad nº 1

1. **Matrices:**
   \[
   A = \begin{pmatrix} 2 & -1 \\ 3 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 4 & -2 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 3 & 5 \\ 2 & -1 & 1 \end{pmatrix}
   \]
   - **a) \( A + B \)**: Possible because both \( A \) and \( B \) are 2x2 matrices.
   - **b) \( AC \)**: Possible because \( A \) is 2x2 and \( C \) is 2x3, so the resulting matrix will be 2x3.
   - **c) \( CB \) and \( C^T B \)**: 
     - \( CB \) is not possible since \( C \) is 2x3 and \( B \) is 2x2.
     - \( C^T \) is the transpose of \( C \) (3x2), so \( C^T B \) is possible (3x2 multiplied by 2x2, resulting in a 3x2 matrix).
   - **d) \( (2A + B)C \)**: Possible, since \( 2A + B \) is 2x2 and \( C \) is 2x3, resulting in a 2x3 matrix.

### Actividad nº 2

2. **Matrices:**
   \[
   A = \begin{pmatrix} 2 & -1 \\ 3 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 4 & -2 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 3 & 5 \\ 2 & -1 & 1 \end{pmatrix}
   \]
   - **a) \( ABC \)**: Not possible because \( A \times B \) results in a 2x2 matrix, but \( BC \) is not defined (since \( B \) is 2x2 and \( C \) is 2x3).
   - **b) \( C^T \left(\frac{1}{2} B - A\right) \)**: 
     - First, calculate \( \frac{1}{2} B \) and then \( \frac{1}{2} B - A \), which is a 2x2 matrix.
     - Multiply by \( C^T \) (3x2), resulting in a 3x2 matrix.
   - **c) \( A^2, B^2, \) and \( C^2 \)**:
     - \( A^2 \) is possible (2x2).
     - \( B^2 \) is possible (2x2).
     - \( C^2 \) is not possible since \( C \) is 2x3.

### Actividad nº 3

3. **Matrices:**
   \[
   A = \begin{pmatrix} 2 & 6 & 3 \\ 0 & 9 & 5 \\ -6 & 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 1 \\ 2 & 4 \\ -2 & 7 \end{pmatrix}
   \]
   - **a) \( AB \) and \( BA \)**: 
     - \( AB \): 3x3 multiplied by 3x2, resulting in a 3x2 matrix.
     - \( BA \): 2x3 multiplied by 3x3, resulting in a 2x3 matrix.
     - The results do not coincide as they have different dimensions.
   - **b) Calculate \( (A + B)^2 \) and \( A^2 + 2AB + B^2 \)**:
     - Check if the results match.
   - **c) Calculate \( A^2 - B^2 \) and \( (A + B)(A - B) \)**:
     - Verify if these results are the same.

Would you like to see the step-by-step calculations for any of these problems?

### Questions
1. Would you like to calculate the matrix additions or multiplications for any specific part?
2. Do you need the detailed step-by-step solution for finding \( C^T \left(\frac{1}{2} B - A\right) \)?
3. Would you like to know the determinant or inverse of any matrix given?
4. Do you need help verifying whether the matrix products \( AB \) and \( BA \) match?
5. Are you interested in a detailed explanation of any specific property of these matrices?

### Tip
When multiplying matrices, always check the dimensions first. If the number of columns in the first matrix matches the number of rows in the second matrix, the multiplication is defined.

Resolver matrices, dadas las matrices A, B y C en el documento proporcionado.

Solve matrix operations including addition, multiplication, and transpose for matrices A, B, and C. This problem covers essential matrix operations, properties of matrix transposition, and matrix exponentiation, suitable for Grades 10-12.

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