If a specific $n$ by $n$ matrix A satisfies the relation that $$A^n=0$$
What information can we get about its rank?
Update: Certainly it is not full rank, but can we dig more out of it?
2026-03-26 06:28:30.1774506510
Rank of a special matrix
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aCTOH's brief comment gives the answer without getting into how. Here is a description.
Take the $n\times n$ identity matrix. Now remove its last column and as replacement put the zero vector, but as the first column. Call this matrix $A_1$.
For any $k<n$ remove the last $k$ columns from $I_n$ and replace them by equal number of zero vectors, but at the front. Call this matrix $A_k$
Note that for all these matrices the rank is the number of non-zero columns, and compute the powers for all these matrices.