Diagonalizability and Repeated Eigenvalues
Consider the matrix \(A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}\). (a) Find the eigenvalues of \(A\) by solving its characteristic equation. (b) Determine whether \(A\) is diagonalizable, and explain how the multiplicity of each eigenvalue influences your conclusion.
Eigenvalue Identification in Structured Matrices
Consider a 3×3 matrix \(M\) where every diagonal entry is \(r\) and every off-diagonal entry is \(c\). Determine all eigenvalues of \(M\) in terms of \(r\) and \(c\).