Let's solve the given eigenvalue problem step by step: ### Step 1: Write the equation in standard eigenvalue form The equation is: \[ \omega^2 \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 1 \end{bmatrix} \{X\} = \begin{bmatrix} 10 & -4 & 0 \\ -4 & 6 & -2 \\ 0 & -2 & 2 \end{bmatrix} \{X\} \] Rewriting: \[ \begin{bmatrix} 10 & -4 & 0 \\ -4 & 6 & -2 \\ 0 & -2 & 2 \end{bmatrix} \{X\} = \omega^2 \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 1 \end{bmatrix} \{X\} \] Move all terms to one side: \[ \left( \begin{bmatrix} 10 & -4 & 0 \\ -4 & 6 & -2 \\ 0 & -2 & 2 \end{bmatrix} - \omega^2 \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 1 \end{bmatrix} \right)\{X\} = 0 \] **Explanation:** We have rewritten the problem in the form of a generalized eigenvalue problem: \( [A] - \omega^2 [B] \). --- ### Step 2: Set up the characteristic equation The equation for non-trivial solutions is: \[ \det\left( \begin{bmatrix} 10 & -4 & 0 \\ -4 & 6 & -2 \\ 0 & -2 & 2 \end{bmatrix} - \omega^2 \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 1 \end{bmatrix} \right) = 0 \] Compute the matrix: \[ \begin{bmatrix} 10 - 3\omega^2 & -4 & 0 \\ -4 & 6 - 2\omega^2 & -2 - 2\omega^2 \\ 0 & -2 & 2 - \omega^2 \end{bmatrix} \] **Explanation:** We substitute and subtract the matrices, forming a determinant equation whose roots will give us the eigenvalues (\(\omega^2\)). --- ### Step 3: Solve the determinant for eigenvalues \(\omega^2\) Compute the determinant: \[ \left| \begin{array}{ccc} 10 - 3\omega^2 & -4 & 0 \\ -4 & 6 - 2\omega^2 & -2 - 2\omega^2 \\ 0 & -2 & 2 - \omega^2 \end{array} \right| = 0 \] Expand along the first row: \[ (10 - 3\omega^2) \left| \begin{array}{cc} 6 - 2\omega^2 & -2 - 2\omega^2 \\ -2 & 2 - \omega^2 \end{array} \right| - (-4) \left| \begin{array}{cc} -4 & -2 - 2\omega^2 \\ 0 & 2 - \omega^2 \end{array} \right| = 0 \] Let's compute: - First minor: \((6 - 2\omega^2)(2 - \omega^2) - (-2 - 2\omega^2)(-2)\) \(= (6 - 2\omega^2)(2 - \omega^2) - 2(2 + 2\omega^2)\) \(= [12 - 6\omega^2 - 4\omega^2 + 2\omega^4] - [4 + 4\omega^2]\) \(= 12 - 10\omega^2 + 2\omega^4 - 4 - 4\omega^2\) \(= 8 - 14\omega^2 + 2\omega^4\) - Second minor: \((-4)[(-4)(2 - \omega^2) - 0(-2 - 2\omega^2)]\) \(= -4 [-4(2 - \omega^2)]\) \(= -4(-8 + 4\omega^2)\) \(= 32 - 16\omega^2\) Thus, the determinant simplifies to: \[ (10 - 3\omega^2)[8 - 14\omega^2 + 2\omega^4] + 32 - 16\omega^2 = 0 \] Expand: \[ (10)(8 - 14\omega^2 + 2\omega^4) - 3\omega^2(8 - 14\omega^2 + 2\omega^4) + 32 - 16\omega^2 = 0 \] \[ 80 - 140\omega^2 + 20\omega^4 - 24\omega^2 + 42\omega^4 - 6\omega^6 + 32 - 16\omega^2 = 0 \] Combine like terms: - Constants: \(80 + 32 = 112\) - \(\omega^2\): \(-140 - 24 - 16 = -180\) - \(\omega^4\): \(20 + 42 = 62\) - \(\omega^6\): \(-6\) So, \[ -6\omega^6 + 62\omega^4 - 180\omega^2 + 112 = 0 \] Divide by -2: \[ 3\omega^6 - 31\omega^4 + 90\omega^2 - 56 = 0 \] **Explanation:** We have formed and simplified the characteristic polynomial equation. The solutions to this equation are the eigenvalues \(\omega^2\). --- ## **Final Answer** **The eigenvalue problem reduces to solving:** \[ 3\omega^6 - 31\omega^4 + 90\omega^2 - 56 = 0 \] **The values of \(\omega^2\) that satisfy this equation are the required eigenvalues.** (You can now solve this cubic equation in \(\omega^2\) using analytical or numerical methods.)

Let's break down the solution step-by-step: ### **Step 1: Formulate the eigenvalue problem** The given matrices are: \[ \mathbf{K} = \begin{bmatrix} 10 & -4 & 0 \\ -4 & 6 & -2 \\ 0 & -2 & 2 \end{bmatrix}, \quad \mathbf{M} = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 2 \\ 0 & 0 & 1 \end{bmatrix} \] The eigenvalue problem is: \[ \mathbf{K} \mathbf{X} = \omega^2 \mathbf{M} \mathbf{X} \] or equivalently, \[ (\mathbf{K} - \omega^2 \mathbf{M}) \mathbf{X} = 0 \] **Explanation:** This is a generalized eigenvalue problem where \(\omega^2\) are the eigenvalues, and \(\mathbf{X}\) are the eigenvectors. --- ### **Step 2: Set up the characteristic equation** The eigenvalues \(\omega^2\) satisfy: \[ \det(\mathbf{K} - \omega^2 \mathbf{M}) = 0 \] Compute the matrix: \[ \mathbf{K} - \omega^2 \mathbf{M} = \begin{bmatrix} 10 - 3 \omega^2 & -4 & 0 \\ -4 & 6 - 2 \omega^2 & -2 - 2 \omega^2 \\ 0 & -2 & 2 - \omega^2 \end{bmatrix} \] **Explanation:** Subtracting \(\omega^2\) times \(\mathbf{M}\) from \(\mathbf{K}\) gives the matrix whose determinant yields the characteristic equation. --- ### **Step 3: Find the eigenvalues by solving the determinant** The determinant: \[ \det(\mathbf{K} - \omega^2 \mathbf{M}) = 0 \] Expanding along the first row: \[ (10 - 3 \omega^2) \times \begin{vmatrix} 6 - 2 \omega^2 & -2 - 2 \omega^2 \\ -2 & 2 - \omega^2 \end{vmatrix} - (-4) \times \begin{vmatrix} -4 & -2 - 2 \omega^2 \\ 0 & 2 - \omega^2 \end{vmatrix} = 0 \] Calculating minors: - First minor: \[ (6 - 2 \omega^2)(2 - \omega^2) - (-2 - 2 \omega^2)(-2) \] \[ = (12 - 6 \omega^2 - 4 \omega^2 + 2 \omega^4) - [2 + 4 \omega^2] \] \[ = (12 - 10 \omega^2 + 2 \omega^4) - (2 + 4 \omega^2) = 10 - 14 \omega^2 + 2 \omega^4 \] - Second minor: \[ -4 \times [( -4)(2 - \omega^2) - 0 \times (-2 - 2 \omega^2)] = -4 \times (-4)(2 - \omega^2) = 16 (2 - \omega^2) = 32 - 16 \omega^2 \] Putting it all together: \[ (10 - 3 \omega^2)(10 - 14 \omega^2 + 2 \omega^4) + 32 - 16 \omega^2 = 0 \] Expanding: \[ (10)(10 - 14 \omega^2 + 2 \omega^4) - 3 \omega^2 (10 - 14 \omega^2 + 2 \omega^4) + 32 - 16 \omega^2 = 0 \] which simplifies to: \[ 100 - 140 \omega^2 + 20 \omega^4 - 30 \omega^2 + 42 \omega^4 - 6 \omega^6 + 32 - 16 \omega^2 = 0 \] Combine like terms: \[ (100 + 32) + (-140 - 30 - 16) \omega^2 + (20 + 42) \omega^4 - 6 \omega^6 = 0 \] \[ 132 - 186 \omega^2 + 62 \omega^4 - 6 \omega^6 = 0 \] Divide through by -2: \[ 3 \omega^6 - 31 \omega^4 + 93 \omega^2 - 66 = 0 \] **Final polynomial in \(\omega^2\):** \[ 3 \omega^6 - 31 \omega^4 + 93 \omega^2 - 66 = 0 \] --- ### **Final answer:** **The eigenvalues \(\omega^2\) are solutions to the polynomial:** \[ \boxed{ 3 \omega^6 - 31 \omega^4 + 93 \omega^2 - 66 = 0 } \] You can solve this degree-6 polynomial (or reduce it to a cubic in \(\lambda = \omega^2\)) using numerical methods or factorization techniques to find the specific eigenvalues \(\omega\).