Discussion
Let \(M = \begin{bmatrix} 4 & -3 \\ 1 & 0 \end{bmatrix}\).
Consider the following statements:
P: \(M^{8} + M^{12}\) is diagonalizable.
Q: \(M^{7} + M^{9}\) is diagonalizable.
Which of the following statements is correct?
Answer: D
Both P and Q are TRUE.
The characteristic polynomial of M is
\[ \lambda^{2} - 4\lambda + 3 = (\lambda-1)(\lambda-3), \]
so M has the distinct eigenvalues 1 and 3 and is diagonalizable: \(M = PDP^{-1}\) with D = diag(1, 3). Any polynomial in M is then \(p(M) = P\,p(D)\,P^{-1}\), which is diagonal in the same basis. Both \(M^{8}+M^{12}\) and \(M^{7}+M^{9}\) are polynomials in M, so both are diagonalizable.
The options declaring either sum non-diagonalizable overlook that powers and sums of a diagonalizable matrix share its eigenvectors.
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