#### Vol. 9, No. 2, 2016

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The kernel of the matrix $\lbrack i\mskip-2mu j \pmod n\rbrack$ when $n$ is prime

### Maria I. Bueno, Susana Furtado, Jennifer Karkoska, Kyanne Mayfield, Robert Samalis and Adam Telatovich

Vol. 9 (2016), No. 2, 265–280
##### Abstract

In this paper, we consider the $n×n$ matrix whose $\left(i,j\right)$-th entry is $ij\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}n\right)$ and compute its rank and a basis for its kernel (viewed as a matrix over the real numbers) when $n$ is prime. We also give a conjecture on the rank of this matrix when $n$ is not prime and give a set of vectors in its kernel, which is a basis if the conjecture is true. Finally, we include an application of this problem to number theory.

##### Keywords
rank of a matrix, kernel of a matrix, bisymmetric matrix.
##### Mathematical Subject Classification 2010
Primary: 15A03, 11M06, 11M20