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The problem involves solving a system of linear equations for a system of three masses suspended by springs. We need to express the equations in matrix form \( \mathbf{A} \mathbf{x} = \mathbf{b} \) and then discuss an approach to solve for \( \mathbf{x} \) and \( \mathbf{A}^{-1} \).

### Given System of Equations

The steady-state equations provided are:

1. \( 3k x_1 - 2k x_2 = m_1 g \)
2. \( -2k x_1 + 3k x_2 - k x_3 = m_2 g \)
3. \( -k x_2 + k x_3 = m_3 g \)

where:
- \( x_1 \), \( x_2 \), and \( x_3 \) are the displacements of the masses \( m_1 \), \( m_2 \), and \( m_3 \), respectively.
- \( k \) is the spring constant.
- \( g \) is the gravitational acceleration.

### Matrix Form \( \mathbf{A} \mathbf{x} = \mathbf{b} \)

The matrix equation \( \mathbf{A} \mathbf{x} = \mathbf{b} \) can be set up as follows:

1. Define the displacement vector \( \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \).
2. Define the constant vector \( \mathbf{b} = \begin{bmatrix} m_1 g \\ m_2 g \\ m_3 g \end{bmatrix} \).
3. The coefficient matrix \( \mathbf{A} \) will capture the coefficients of \( x_1 \), \( x_2 \), and \( x_3 \) from each equation:

\[
\mathbf{A} = \begin{bmatrix} 3k & -2k & 0 \\ -2k & 3k & -k \\ 0 & -k & k \end{bmatrix}
\]

Thus, the system can be written as:

\[
\begin{bmatrix} 3k & -2k & 0 \\ -2k & 3k & -k \\ 0 & -k & k \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} m_1 g \\ m_2 g \\ m_3 g \end{bmatrix}
\]

### Solving for \( \mathbf{x} \) and \( \mathbf{A}^{-1} \)

1. **Finding \( \mathbf{x} \):**
   - To solve for \( \mathbf{x} \), compute \( \mathbf{x} = \mathbf{A}^{-1} \mathbf{b} \).
   - If \( \mathbf{A} \) is invertible, \( \mathbf{A}^{-1} \) can be found using methods like Gaussian elimination, matrix row reduction, or directly calculating the inverse if feasible.
   - After finding \( \mathbf{A}^{-1} \), multiply it by \( \mathbf{b} \) to get \( \mathbf{x} \).

2. **Finding \( \mathbf{A}^{-1} \):**
   - To find \( \mathbf{A}^{-1} \), if it exists, calculate it using the formula for the inverse of a 3x3 matrix:
     \[
     \mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \text{adj}(\mathbf{A})
     \]
   - Compute the determinant \( \det(\mathbf{A}) \) and the adjugate matrix \( \text{adj}(\mathbf{A}) \).
   - If \( \det(\mathbf{A}) \neq 0 \), then \( \mathbf{A}^{-1} \) exists and can be used to solve for \( \mathbf{x} \).

Would you like further details on calculating the determinant and adjugate for this specific matrix? 

### Follow-up Questions

1. How can we interpret the physical meaning of the solutions for \( x_1 \), \( x_2 \), and \( x_3 \)?
2. What happens if \( \det(\mathbf{A}) = 0 \) in terms of the system's behavior?
3. How do different values of \( k \) and \( m \) impact the stability of the system?
4. What are alternative methods to solve the matrix equation if \( \mathbf{A} \) is not invertible?
5. How can we extend this approach to systems with more than three masses?

**Tip:** For matrix inversion, if the matrix is symmetric (like in some spring systems), specialized algorithms can optimize the calculation.

Express the system of equations for three masses suspended by springs in matrix form \( \mathbf{A} \mathbf{x} = \mathbf{b} \) and discuss an approach to solve for \( \mathbf{x} \) and \( \mathbf{A}^{-1} \).

Explore the method of solving a system of equations for a three-mass spring system using matrix equations. Learn how to express the equations in the form \( \mathbf{A} \mathbf{x} = \mathbf{b} \), apply Newton’s second law and Hooke’s law, and compute the inverse matrix \( \mathbf{A}^{-1} \) to find the displacements. Ideal for engineering or physics students.

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