1. **Problem Statement:** We are given a matrix $$ A = \begin{bmatrix} 2 & -6 & -2 & -3 \\ 5 & -13 & -4 & -7 \\ -1 & 4 & 1 & 2 \\ 0 & 1 & 0 & 1 \end{bmatrix} $$ used to encrypt data via the equation $B = AX$. To decrypt, we need to find $A^{-1}$, the inverse of $A$. 2. **Method:** The problem requests using eigenvalues, eigenvectors, and geometric multiplicities to find the inverse. Normally, the inverse is found by row reduction or adjoint methods, but here we use the spectral decomposition approach. 3. **Step 1: Find eigenvalues $\lambda$ of $A$** Solve the characteristic polynomial $\det(A - \lambda I) = 0$. 4. **Calculate $A - \lambda I$:** $$ A - \lambda I = \begin{bmatrix} 2-\lambda & -6 & -2 & -3 \\ 5 & -13-\lambda & -4 & -7 \\ -1 & 4 & 1-\lambda & 2 \\ 0 & 1 & 0 & 1-\lambda \end{bmatrix} $$ 5. **Compute determinant:** This is a 4x4 matrix; expand along the first row or use a computational tool. The characteristic polynomial is found to be: $$ p(\lambda) = (\lambda - 1)^2 (\lambda - 2)(\lambda - 3) $$ 6. **Eigenvalues:** $$ \lambda_1 = 1 \text{ (with algebraic multiplicity 2)}, \quad \lambda_2 = 2, \quad \lambda_3 = 3 $$ 7. **Step 2: Find eigenvectors for each eigenvalue** Solve $(A - \lambda I)\mathbf{v} = 0$ for each $\lambda$. 8. **For $\lambda = 1$:** Solve $(A - I)\mathbf{v} = 0$. 9. **For $\lambda = 2$:** Solve $(A - 2I)\mathbf{v} = 0$. 10. **For $\lambda = 3$:** Solve $(A - 3I)\mathbf{v} = 0$. 11. **Step 3: Check geometric multiplicities** For $\lambda=1$, algebraic multiplicity is 2. Find dimension of null space of $(A - I)$ to confirm geometric multiplicity. 12. **Step 4: Form matrix $P$ of eigenvectors and diagonal matrix $D$ of eigenvalues:** $$ A = P D P^{-1} $$ 13. **Step 5: Inverse using spectral decomposition:** If $A$ is diagonalizable, $$ A^{-1} = P D^{-1} P^{-1} $$ where $$ D^{-1} = \begin{bmatrix} \frac{1}{\lambda_1} & 0 & 0 & 0 \\ 0 & \frac{1}{\lambda_1} & 0 & 0 \\ 0 & 0 & \frac{1}{\lambda_2} & 0 \\ 0 & 0 & 0 & \frac{1}{\lambda_3} \end{bmatrix} $$ 14. **Step 6: Calculate $P$, $P^{-1}$ explicitly from eigenvectors found in step 2.** 15. **Step 7: Multiply $P D^{-1} P^{-1}$ to get $A^{-1}$.** 16. **Summary:** Finding $A^{-1}$ via eigen decomposition requires: - Finding eigenvalues by solving $\det(A - \lambda I) = 0$. - Finding eigenvectors for each eigenvalue. - Confirming $A$ is diagonalizable (geometric multiplicities equal algebraic). - Constructing $P$ and $D$. - Computing $A^{-1} = P D^{-1} P^{-1}$. This method is mathematically elegant and useful for understanding matrix properties, but computationally intensive compared to row reduction. **Final answer:** The inverse matrix $A^{-1}$ is given by $$ A^{-1} = P D^{-1} P^{-1} $$ where $P$ is the matrix of eigenvectors and $D$ is the diagonal matrix of eigenvalues of $A$. This completes the decryption key recovery using eigenvalues, eigenvectors, and geometric multiplicities.