https://file.aiquickdraw.com/mathbot/file/17239779321696vfndfcu.jpg

\[

    \begin{bmatrix}

        1 & -\frac{1}{2} & \frac{1}{\sqrt{2}} \\

        2 & \frac{3}{2}  & 0                  \\

        0 & \frac{5}{2}  & \sqrt{2}

    \end{bmatrix}

    \begin{bmatrix}

        F_1 \\

        F_2 \\

        F_3

    \end{bmatrix}

    =

    \begin{bmatrix}

        0 \\

        0 \\

        0

    \end{bmatrix}

\]

The problem in the image describes a system of three equations with three unknown forces \( F_1 \), \( F_2 \), and \( F_3 \). The student is asked if they can solve for \( F_1 \), \( F_2 \), and \( F_3 \) uniquely from these equations.

The system of equations can be written in matrix form as:

\[
\begin{bmatrix}
1 & -\frac{1}{2} & \frac{1}{\sqrt{2}} \\
2 & \frac{3}{2} & 0 \\
0 & \frac{5}{2} & \sqrt{2}
\end{bmatrix}
\begin{bmatrix}
F_1 \\
F_2 \\
F_3
\end{bmatrix}
=
\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix}
\]

This is a homogeneous system of linear equations, and the key question is whether this system has a unique solution (which would be the trivial solution \( F_1 = F_2 = F_3 = 0 \)) or if there are non-trivial solutions.

To determine if the system has a unique solution, we can check the determinant of the coefficient matrix. If the determinant is non-zero, the system has a unique solution. If the determinant is zero, then the system has infinitely many solutions.

Let me calculate the determinant of the coefficient matrix.import numpy as np

# Defining the coefficient matrix
A = np.array([
    [1, -1/2, 1/np.sqrt(2)],
    [2, 3/2, 0],
    [0, 5/2, np.sqrt(2)]
])

# Calculating the determinant of the matrix
determinant = np.linalg.det(A)
determinantThe determinant of the coefficient matrix is approximately \( 7.07 \), which is non-zero. 

Since the determinant is non-zero, the system of equations has a unique solution. In this case, the unique solution for the homogeneous system is the trivial solution, where \( F_1 = F_2 = F_3 = 0 \).

Therefore, the student can indeed solve for \( F_1 \), \( F_2 \), and \( F_3 \) uniquely, and the only solution is the trivial one.

Would you like more details or have any questions about this?

Here are some related questions:
1. What is a homogeneous system of equations?
2. Why is the determinant important for solving systems of equations?
3. How can non-trivial solutions arise in linear systems?
4. What is the geometric interpretation of a determinant in 3D space?
5. How does one compute a determinant for larger matrices?

**Tip:** When solving a homogeneous system of equations, always check the determinant first to determine the nature of the solutions (trivial or non-trivial).

Learn how to solve a homogeneous system of linear equations using determinants. This problem involves determining if a system has a unique solution through the calculation of the determinant. Suitable for high school students studying linear algebra.

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