Nilpotents Leave No Trace: A Matrix Mystery for Pandemic Times

Abstract

Reopening a cold case, inspector Echelon, high-ranking in the Row Operations Center, is searching for a lost linear map, known to be nilpotent. When a partially decomposed matrix is unearthed, he reconstructs its reduced form, finding it singular. But were its roots nilpotent? 1. Early In the Investigation In teaching Linear Algebra, the first topic often is row reduction [1, 7], including Row Reduced Echelon Form (RREF); its applicability is broad and growing. Another topic, surprisingly popular with beginning students, is nilpotent matrices. One naturally wonders about their intersection. For instance, one would expect to find a book exercise asking: What can be said about the RREF of a nilpotent matrix? In the early days of the Covid-19 pandemic, as test delivery went remote, demand grew for new, Internet-resistant problems. A limited literature search for the Nilpotent-RREF connection came up short, suggesting potential for take-home final exam questions, hence the note at hand. We’ll first explore examples sufficient to settle the 3×3 case, then consider the general situation. The upshot is the row reduction eliminates all traces of nilpotence. 2. Stumbling On Evidence We refer to [1, 2] for general background on RREF and rank. Recall that a square matrix M is nilpotent if some power of M , say M, is the zero matrix; the smallest such k is called the niloptent index or just index of M . For instance, the rightmost matrix in (2) below is nilpotent, of index 3. Indeed, every strictly upper-triangular matrix (square, with zeros on