1. **Stating the problem:** We are given a system of linear equations in matrix form: $$\begin{bmatrix} \theta_1 \\ v_1 \\ \theta_2 \\ v_2 \end{bmatrix} = \begin{bmatrix} 833.54 & -5001.27 & 416.77 & ? \\ -5001.27 & 80020.32 & 5001.27 & ? \\ 416.77 & 5001.27 & 1667.09 & ? \\ 5001.27 & -5001.27 & 0 & ? \end{bmatrix} \begin{bmatrix} 0 \\ 10 \\ 0 \\ 0 \end{bmatrix}$$ (Note: The original matrix is 4x4 but the last column is missing in the problem statement; assuming the last column is zeros as per the load vector size.) We want to find the displacement vector $$\begin{bmatrix} \theta_1 & v_1 & \theta_2 & v_2 \end{bmatrix}^T$$ by calculating the inverse of the stiffness matrix and multiplying it by the load vector. 2. **Formula used:** $$\mathbf{x} = \mathbf{K}^{-1} \mathbf{F}$$ where $$\mathbf{K}$$ is the stiffness matrix, $$\mathbf{F}$$ is the load vector, and $$\mathbf{x}$$ is the displacement vector. 3. **Important rules:** - The inverse of a matrix $$\mathbf{K}$$ exists only if $$\det(\mathbf{K}) \neq 0$$. - The inverse $$\mathbf{K}^{-1}$$ satisfies $$\mathbf{K} \mathbf{K}^{-1} = \mathbf{I}$$, where $$\mathbf{I}$$ is the identity matrix. - For a 4x4 matrix, the inverse can be computed using Gaussian elimination or adjoint method, but it is lengthy. 4. **Step-by-step solution:** - Write the stiffness matrix $$\mathbf{K}$$ explicitly: $$\mathbf{K} = \begin{bmatrix} 833.54 & -5001.27 & 416.77 & 0 \\ -5001.27 & 80020.32 & 5001.27 & 0 \\ 416.77 & 5001.27 & 1667.09 & 0 \\ 5001.27 & -5001.27 & 0 & 0 \end{bmatrix}$$ - Write the load vector $$\mathbf{F} = \begin{bmatrix} 0 \\ 10 \\ 0 \\ 0 \end{bmatrix}$$ - Since the last column and last row have zeros except some entries, the matrix is singular or near singular. We check determinant: 5. **Determinant check (conceptual):** - Because the last row has zeros except the first two elements, and the last column is zeros, the determinant is zero or very close to zero, so inverse may not exist or is ill-conditioned. 6. **Alternative approach:** - Since the last column is zero, the system is underdetermined or singular. - We can solve the system $$\mathbf{K} \mathbf{x} = \mathbf{F}$$ directly using Gaussian elimination or pseudo-inverse. 7. **Solving the system:** - Write the system explicitly: $$\begin{cases} 833.54 \theta_1 - 5001.27 v_1 + 416.77 \theta_2 + 0 \cdot v_2 = 0 \\ -5001.27 \theta_1 + 80020.32 v_1 + 5001.27 \theta_2 + 0 \cdot v_2 = 10 \\ 416.77 \theta_1 + 5001.27 v_1 + 1667.09 \theta_2 + 0 \cdot v_2 = 0 \\ 5001.27 \theta_1 - 5001.27 v_1 + 0 \cdot \theta_2 + 0 \cdot v_2 = 0 \end{cases}$$ - From the last equation: $$5001.27 \theta_1 - 5001.27 v_1 = 0 \implies \theta_1 = v_1$$ - Substitute $$\theta_1 = v_1$$ into the other equations: First equation: $$833.54 v_1 - 5001.27 v_1 + 416.77 \theta_2 = 0 \implies (-4167.73) v_1 + 416.77 \theta_2 = 0$$ Second equation: $$-5001.27 v_1 + 80020.32 v_1 + 5001.27 \theta_2 = 10 \implies 75019.05 v_1 + 5001.27 \theta_2 = 10$$ Third equation: $$416.77 v_1 + 5001.27 v_1 + 1667.09 \theta_2 = 0 \implies 5418.04 v_1 + 1667.09 \theta_2 = 0$$ - Now solve the system: $$\begin{cases} -4167.73 v_1 + 416.77 \theta_2 = 0 \\ 75019.05 v_1 + 5001.27 \theta_2 = 10 \\ 5418.04 v_1 + 1667.09 \theta_2 = 0 \end{cases}$$ - From the first equation: $$416.77 \theta_2 = 4167.73 v_1 \implies \theta_2 = \frac{4167.73}{416.77} v_1 \approx 10 v_1$$ - From the third equation: $$5418.04 v_1 + 1667.09 \theta_2 = 0$$ Substitute $$\theta_2 = 10 v_1$$: $$5418.04 v_1 + 1667.09 \times 10 v_1 = 0 \implies (5418.04 + 16670.9) v_1 = 0 \implies 22088.94 v_1 = 0 \implies v_1 = 0$$ - Then $$\theta_2 = 10 \times 0 = 0$$ - Substitute $$v_1 = 0$$ and $$\theta_2 = 0$$ into the second equation: $$75019.05 \times 0 + 5001.27 \times 0 = 10 \implies 0 = 10$$ - This is a contradiction, so no solution exists for the given system with the assumed matrix. **Conclusion:** The stiffness matrix is singular or inconsistent with the load vector, so the inverse does not exist or the system has no solution. **Final answer:** The inverse of the given stiffness matrix does not exist or the system is unsolvable with the given load vector.