A real 7×7 M has characteristic polynomial (x−1)^α(x−2)^β(x−3)², α>β. Each of M−I, M−2I, M−3I has rank 5. Find m_M(5), where m_M is the minimal polynomial (integer).
Each eigenspace has dimension 7−5=2, so α,β≥2 and α+β=5 force α=3,β=2. Eigenvalue 1 has two Jordan blocks totaling three: sizes 2,1. Eigenvalues 2 and 3 each have two size-one blocks. Minimal polynomial is (x−1)²(x−2)(x−3). At x=5 it is 4²×3×2 = **96**.
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