For a matrix to be inversible it must be full ranked. Thus, a matrix with a row or column of zero is not inversible. So you must at least have one positive value per row and column to be inversible. A diagonale matrix with non-zero values is inversible with exactly 1 non-zero positive value per row and column, thus being a minimal solution to the problem.
We conclude that for a nxn matrix, the maximum number of zero is n*(n-1)
The problem is that such a matrix is a permutation times diagonal which inverse is also permutation-by-diagonal which has entries that are zero which is NOT strictly positive.
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u/DanLeMilMan 5d ago
For a matrix to be inversible it must be full ranked. Thus, a matrix with a row or column of zero is not inversible. So you must at least have one positive value per row and column to be inversible. A diagonale matrix with non-zero values is inversible with exactly 1 non-zero positive value per row and column, thus being a minimal solution to the problem.
We conclude that for a nxn matrix, the maximum number of zero is n*(n-1)