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Solve for FA, FB, and FC within the matrix

To solve for \( F_A \), \( F_B \), and \( F_C \) from the matrix equation, we will treat the problem as a system of linear equations in matrix form:

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

This system of equations can be written as three separate linear equations:

1. \( 0.371 F_A - 0.371 F_C = -310.56 \)
2. \( 0.928 F_A + F_B + 0.928 F_C = 1159.08 \)
3. \( 3 F_B + 5.939 F_C = 4492.08 \)

We will solve this step by step:

### Step 1: Solve for \( F_A \) and \( F_C \) from Equation 1
From equation (1):
\[
0.371 F_A - 0.371 F_C = -310.56
\]

Dividing by 0.371:
\[
F_A - F_C = -837.02
\]
\[
F_A = F_C - 837.02 \quad (4)
\]

### Step 2: Use Equation (4) in Equation 2
Substitute \( F_A = F_C - 837.02 \) into equation (2):
\[
0.928 (F_C - 837.02) + F_B + 0.928 F_C = 1159.08
\]
Expanding:
\[
0.928 F_C - 776.99 + F_B + 0.928 F_C = 1159.08
\]
\[
1.856 F_C + F_B = 1159.08 + 776.99
\]
\[
1.856 F_C + F_B = 1936.07 \quad (5)
\]

### Step 3: Solve Equation 3 for \( F_B \)
From equation (3):
\[
3 F_B + 5.939 F_C = 4492.08
\]
Solve for \( F_B \):
\[
F_B = \frac{4492.08 - 5.939 F_C}{3} \quad (6)
\]

### Step 4: Substitute Equation (6) into Equation (5)
Substitute \( F_B \) from equation (6) into equation (5):
\[
1.856 F_C + \frac{4492.08 - 5.939 F_C}{3} = 1936.07
\]
Multiply everything by 3 to clear the denominator:
\[
3(1.856 F_C) + (4492.08 - 5.939 F_C) = 3(1936.07)
\]
\[
5.568 F_C + 4492.08 - 5.939 F_C = 5808.21
\]
Simplify:
\[
-0.371 F_C + 4492.08 = 5808.21
\]
\[
-0.371 F_C = 5808.21 - 4492.08
\]
\[
-0.371 F_C = 1316.13
\]
\[
F_C = \frac{1316.13}{-0.371}
\]
\[
F_C = -3548.59
\]

### Step 5: Solve for \( F_A \) and \( F_B \)
From equation (4):
\[
F_A = -3548.59 - 837.02
\]
\[
F_A = -4385.61
\]

From equation (6):
\[
F_B = \frac{4492.08 - 5.939(-3548.59)}{3}
\]
\[
F_B = \frac{4492.08 + 21072.72}{3}
\]
\[
F_B = \frac{25564.8}{3}
\]
\[
F_B = 8521.60
\]

### Final Solution:
\[
F_A = -4385.61, \quad F_B = 8521.60, \quad F_C = -3548.59
\]

Would you like further details or explanations? Here are five questions to deepen your understanding:

1. How can this method be applied to larger systems of linear equations?
2. What is the significance of matrix inversion in solving linear systems?
3. How do changes in the coefficients affect the outcome of \( F_A \), \( F_B \), and \( F_C \)?
4. How can you verify the solution using substitution?
5. How does matrix decomposition (LU or QR) assist in solving linear systems?

**Tip:** Always double-check your matrix operations by substituting the solutions back into the original system to verify the accuracy.

Solve for FA, FB, and FC within the matrix.

Learn how to solve a system of linear equations involving a 3x3 matrix to find values for FA, FB, and FC. This problem includes key concepts from linear algebra, such as matrix multiplication and systems of equations. Suitable for advanced high school or college-level students.

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