Is there a relationship to describe the maximum number of non-zero elements for any matrix, A, of size (n,n), such that the matrix product AA is equal to the zero matrix?
Just curious about how to solve this question.
2026-03-25 01:27:13.1774402033
Maximum number of non-zero elements for an (n, n) matrix product
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The maximum number is in fact $n^2$, i.e., all coefficients can be non-zero. Consider $$ A=\begin{pmatrix} 1 & -1 \cr 1 & -1 \end{pmatrix}. $$ Then $A^2=0$. Another example is $$ A=\begin{pmatrix} 1 & 1 & -2 \cr 1 & 1 & -2 \cr 1 & 1 & -2 \end{pmatrix}. $$ You can generalise this, if you want.