Correct answer: (C) M always has a pair of orthogonal eigenvectors.
Explanation
The nonzero vector v=(−c,1,0) satisfies Mv=0 for every parameter choice. Also w=(0,0,1) satisfies Mw=w. Their dot product is zero. Hence an orthogonal eigenvector pair always exists, C. The displayed zero-eigenvalue vector also refutes full rank.
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