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The problem provided involves solving systems of linear equations with matrices. The matrices given are:

\[
A = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix}, \quad B = \begin{pmatrix}
1 & 4 & 7 \\
2 & 5 & 8 \\
3 & 6 & 9
\end{pmatrix}
\]
and the vector \( b = (1, 4, 7)^T \).

The task is to find the general solution to the following systems:
1. \( A \mathbf{x} = b \)
2. \( A \mathbf{x} = B \mathbf{x} \)
3. \( A \mathbf{x} = B \mathbf{y} \)

Let's break down and analyze each system:

### Part a) \( A \mathbf{x} = b \)
This is a system of linear equations. To solve it, we need to find a vector \( \mathbf{x} \) such that the matrix multiplication \( A \mathbf{x} \) results in \( b \). We can approach this either by calculating the inverse of \( A \) (if it exists) or by using methods like Gaussian elimination or Cramer's rule. However, note that matrix \( A \) has dependent rows, meaning it is singular (non-invertible), so we may need to look for a solution in terms of a parametric form, or determine if no solutions exist.

### Part b) \( A \mathbf{x} = B \mathbf{x} \)
This represents an equation where we must find \( \mathbf{x} \) such that \( A \mathbf{x} = B \mathbf{x} \). The solution involves checking whether both sides are equal and can be solved by finding eigenvalues or by simplifying the equation to express the relationship between \( A \) and \( B \).

### Part c) \( A \mathbf{x} = B \mathbf{y} \)
Here, we have two unknowns: \( \mathbf{x} \) and \( \mathbf{y} \). This system might require us to express \( \mathbf{x} \) and \( \mathbf{y} \) in terms of each other, or look for specific solutions where both matrices act similarly on vectors \( \mathbf{x} \) and \( \mathbf{y} \).

Would you like a detailed step-by-step solution for each part? 

Here are 5 related questions to expand the topic:
1. How do we determine if a matrix is invertible or singular?
2. What methods can be used to solve a system of linear equations involving singular matrices?
3. What is the geometric interpretation of the equation \( A \mathbf{x} = B \mathbf{x} \)?
4. How can we use eigenvalues and eigenvectors to simplify the solution of matrix equations?
5. What is the significance of dependent rows in a matrix?

Tip: When working with matrices, always check the determinant to see if the matrix is invertible before attempting to solve!

Given matrices A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] and B = [[1, 4, 7], [2, 5, 8], [3, 6, 9]], and vector b = (1, 4, 7)^T, find the general solution to the systems: a) Ax = b, b) Ax = Bx, c) Ax = By.

Learn how to solve the systems Ax = b, Ax = Bx, and Ax = By with the matrices A and B provided. This problem covers key concepts of linear algebra, including matrix multiplication and systems of equations, suitable for undergraduate students.

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