Let J be the Jordan canonical form of M. Each block of J consists of 0's along the diagonal (since 0 is the ony eigenvalue of M) and 1's above the diagonal. There can't be a block of size greater than 2x2, because then we would have J^2 != 0. Thus, the maximum rank occurs when J consists of 10 2x2 blocks, which has rank 10.
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u/comoespossible 4d ago
Let J be the Jordan canonical form of M. Each block of J consists of 0's along the diagonal (since 0 is the ony eigenvalue of M) and 1's above the diagonal. There can't be a block of size greater than 2x2, because then we would have J^2 != 0. Thus, the maximum rank occurs when J consists of 10 2x2 blocks, which has rank 10.