True or False: A matrix with two distinct eigenvalues is diagonalizable.

• A. False
• B. Sometimes
• C. True
• D. Only if the eigenvectors are orthogonal

Answer: (C)

The essential idea is that diagonalizability requires a full set of independent eigenvectors that spans the whole space. Having two distinct eigenvalues gives you at least two independent eigenvectors, but that may not be enough to fill an n-dimensional space if n > 2. So the statement is not true in general.

For example, a 3×3 matrix can have eigenvalues 0 (with multiplicity 2) and 1 (with multiplicity 1) and yet only have two linearly independent eigenvectors—in one eigenspace for 0 and one for 1. Such a matrix cannot be diagonalized because you don’t have three independent eigenvectors to form a basis.

However, in the specific case of a 2×2 matrix, two distinct eigenvalues do guarantee diagonalizability, since you would have two independent eigenvectors to form a basis.