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Solve the system of equations for FA, FB, FC using reverse matrix

The system of equations you provided is represented in matrix form as follows:

\[
\begin{bmatrix}
0.371 & 0 & -0.371 \\
0.928 & 1 & 0.928 \\
0 & 3 & 5.939
\end{bmatrix}
\begin{bmatrix}
F_A \\
F_B \\
F_C
\end{bmatrix}
=
\begin{bmatrix}
-310.56 \\
1159.08 \\
4492.08
\end{bmatrix}
\]

To solve this using matrix inversion, the system can be written as:

\[
A \mathbf{F} = \mathbf{B}
\]

where \( A \) is the coefficient matrix, \( \mathbf{F} \) is the force vector \([F_A, F_B, F_C]^T\), and \( \mathbf{B} \) is the result vector.

We need to find \( \mathbf{F} = A^{-1} \mathbf{B} \), where \( A^{-1} \) is the inverse of the matrix \( A \).

### Steps:
1. **Find the inverse of matrix A.**
2. **Multiply the inverse of A by the vector B to get the values of \( F_A, F_B, F_C \).**

Let me calculate the inverse of \( A \) and solve for \( F_A, F_B, F_C \).The solution for the forces \( F_A \), \( F_B \), and \( F_C \) is:

\[
F_A = -4383.22, \quad F_B = 8517.52, \quad F_C = -3546.13
\]

Would you like further details on how the inverse was calculated, or any specific steps of the process?

Here are some related questions to explore further:

1. What is the importance of matrix inversion in solving linear systems?
2. How does the determinant of a matrix affect the ability to find an inverse?
3. Can all systems of equations be solved using the inverse matrix method?
4. What happens if the matrix is singular (non-invertible)?
5. How do numerical inaccuracies affect matrix inversion in real-world problems?

**Tip:** Always check the determinant of a matrix before attempting to invert it. If the determinant is zero, the matrix does not have an inverse.

Solve the system of equations for FA, FB, FC using reverse matrix.

Learn how to solve a system of linear equations using matrix inversion. The system involves three variables FA, FB, and FC. The process includes finding the inverse of a 3x3 matrix and multiplying it by the result vector. Suitable for students studying linear algebra at the college level.

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