Math Problem Statement
[
\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}
]
Solution
The problem in the image describes a system of three equations with three unknown forces , , and . The student is asked if they can solve for , , and uniquely from these equations.
The system of equations can be written in matrix form as:
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).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear algebra
Homogeneous systems of equations
Determinants
Formulas
-
Theorems
-
Suitable Grade Level
High School
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