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Let’s solve this step-by-step. First, we will calculate the matrix products and sums \( AB \), \( BA \), \( AC \), \( BC \), \( A + B \), and \( B + C \). Then, we'll verify the given properties:

\[
(AB)C = A(BC), \quad (A+B)C = AC + BC, \quad A(B+C) = AB + AC
\]

### Given Matrices:
\[
A = \begin{bmatrix} 2 & 3 & 5 \\ 1 & 0 & -1 \\ -2 & 1 & 4 \end{bmatrix}, \quad
B = \begin{bmatrix} 4 & 2 & 0 \\ 1 & -5 & -2 \\ 0 & 2 & 3 \end{bmatrix}, \quad
C = \begin{bmatrix} -3 & 2 & -1 \\ 5 & 2 & 8 \\ 2 & 3 & -4 \end{bmatrix}.
\]

I'll first compute each expression systematically. Hold on while I calculate these values. numpy as np

# Define matrices A, B, C
A = np.array([[2, 3, 5], [1, 0, -1], [-2, 1, 4]])
B = np.array([[4, 2, 0], [1, -5, -2], [0, 2, 3]])
C = np.array([[-3, 2, -1], [5, 2, 8], [2, 3, -4]])

# Compute required operations
AB = np.dot(A, B)  # AB
BA = np.dot(B, A)  # BA
AC = np.dot(A, C)  # AC
BC = np.dot(B, C)  # BC
A_plus_B = A + B   # A + B
B_plus_C = B + C   # B + C

# Verifying distributive properties
AB_C = np.dot(AB, C)  # (AB)C
A_BC = np.dot(A, np.dot(B, C))  # A(BC)

A_plus_B_C = np.dot(A + B, C)  # (A + B)C
AC_plus_BC = AC + BC  # AC + BC

A_B_plus_C = np.dot(A, B + C)  # A(B + C)
AB_plus_AC = AB + AC  # AB + AC

AB, BA, AC, BC, A_plus_B, B_plus_C, AB_C, A_BC, A_plus_B_C, AC_plus_BC, A_B_plus_C, AB_plus_ACHere are the computed results:

1. **Matrix Products and Sums:**
   - \( AB = \begin{bmatrix} 11 & -1 & 9 \\ 4 & 0 & -3 \\ -7 & -1 & 10 \end{bmatrix} \)
   - \( BA = \begin{bmatrix} 10 & 12 & 18 \\ 1 & 1 & 2 \\ -4 & 3 & 10 \end{bmatrix} \)
   - \( AC = \begin{bmatrix} 19 & 25 & 2 \\ -5 & -1 & 3 \\ 19 & 10 & -6 \end{bmatrix} \)
   - \( BC = \begin{bmatrix} -2 & 12 & 12 \\ -32 & -14 & -33 \\ 16 & 13 & 4 \end{bmatrix} \)
   - \( A + B = \begin{bmatrix} 6 & 5 & 5 \\ 2 & -5 & -3 \\ -2 & 3 & 7 \end{bmatrix} \)
   - \( B + C = \begin{bmatrix} 1 & 4 & -1 \\ 6 & -3 & 6 \\ 2 & 5 & -1 \end{bmatrix} \)

2. **Verification of Properties:**
   - \( (AB)C = A(BC) = \begin{bmatrix} -20 & 47 & -55 \\ -18 & -1 & 8 \\ 36 & 14 & -41 \end{bmatrix} \)  
     \((AB)C = A(BC)\) is verified.
   - \( (A+B)C = AC + BC = \begin{bmatrix} 17 & 37 & 14 \\ -37 & -15 & -30 \\ 35 & 23 & -2 \end{bmatrix} \)  
     \((A+B)C = AC + BC\) is verified.
   - \( A(B+C) = AB + AC = \begin{bmatrix} 30 & 24 & 11 \\ -1 & -1 & 0 \\ 12 & 9 & 4 \end{bmatrix} \)  
     \(A(B+C) = AB + AC\) is verified.

Would you like further clarification or a deeper breakdown of any step?  

### Questions to Explore:
1. What are the basic properties of matrix multiplication that allow such verifications?
2. Why is \( AB \neq BA \) in general for matrices?
3. How do dimensions affect the possibility of computing sums and products of matrices?
4. Can the distributive property be used for other algebraic objects besides matrices?
5. What is the importance of verifying such properties in applications of linear algebra?

### Tip:
Always verify matrix dimensions before attempting multiplication, as the number of columns in the first matrix must equal the number of rows in the second.

Evaluate AB, BA, AC, BC, A+B, B+C, and verify the properties: (AB)C = A(BC), (A+B)C = AC+BC, A(B+C) = AB+AC for the given matrices A, B, and C.

Explore matrix operations including multiplication and addition for the matrices A, B, and C. Verify properties such as (AB)C = A(BC) and (A+B)C = AC+BC. Step-by-step solutions are provided for these key linear algebra concepts.

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