Let's solve the problem in two steps:

1. Matrix Multiplication $A \cdot B$:
   Given matrices $A$ and $B$, we need to compute the product $A \cdot B$.

   Matrix $A$:

   -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}$$ Matrix $$B$$: $$B = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}$$ The product $$A \cdot B$$ is computed as follows: $$(A \cdot B)_{ij} = \sum_{k=1}^3 A_{ik} \cdot B_{kj}$$

2. Solving the system of linear equations: The system is:

   x - y + 3z &= 4 \\ x - 2y - 2z &= 9 \\ 2x + y + 3z &= 1 \end{aligned}$$ This system can be written as a matrix equation $$C \cdot \vec{x} = \vec{b}$$, where: $$C = \begin{bmatrix} 1 & -1 & 3 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}, \quad \vec{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \vec{b} = \begin{bmatrix} 4 \\ 9 \\ 1 \end{bmatrix}$$ Solve for $$\vec{x}$$ using $$\vec{x} = C^{-1} \cdot \vec{b}$$.

Let me compute these step-by-step.import numpy as np

Define matrices A and B

A = np.array([[-4, 4, 4], [-7, 1, 3], [5, -3, -1]]) B = np.array([[1, -1, 1], [1, -2, -2], [2, 1, 3]])

Compute A * B

AB = np.dot(A, B)

Define the coefficient matrix C and vector b for the system of equations

C = np.array([[1, -1, 3], [1, -2, -2], [2, 1, 3]]) b = np.array([4, 9, 1])

Solve the system of equations C * x = b

solution = np.linalg.solve(C, b)

AB, solution(array([[8, 0, 0], [0, 8, 0], [0, 0, 8]]), array([ 2.55555556, -2.77777778, -0.44444444]))(array([[8, 0, 0], [0, 8, 0], [0, 0, 8]]), array([ 2.55555556, -2.77777778, -0.44444444]))successHere are the results:

1. Matrix Multiplication $A \cdot B$: 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{bmatrix}$$
2. Solution to the system of equations: $x = 2.56, \quad y = -2.78, \quad z = -0.44$ (Values are rounded to two decimal places.)

Would you like detailed steps for the matrix multiplication or solving the system of equations?

Here are 5 additional related questions for exploration:

1. What is the determinant of matrix $A$?
2. Can the system of equations be solved using Gaussian elimination? If so, how?
3. What are the eigenvalues and eigenvectors of $A \cdot B$?
4. Is $A \cdot B$ a diagonal matrix? Why?
5. Can matrix $B$ be inverted, and if so, what is its inverse?

Tip: Understanding how to solve linear equations with matrix methods will help in working with systems of equations in applied contexts like engineering and physics!