The non-zero determinant is not divisible by three (Posted on 2023-06-12)

	Submitted by Brian Smith
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If there are two identical rows or two identical columns of of "1 1 1 1 1 1 1 1 1 1", then the determinant is zero. So then for the determinant to be non-zero the nine non-1 entries must all occupy different rows and columns. Use row swaps and/or column swaps to permute those entries to lay on the first nine positions of the main diagonal. Note these operations will only change the sign of the determinant and not affect divisibility. The bottom row will be a row of all 1's. Then add the negative of the bottom row to the other nine rows. These operations will not change the value of the determinant. Now the matrix is a lower-triangular matrix. The entries on the main diagonal are either 2-1=1 or 0-1=-1. Then the determinant is easily evaluated to be the product of these entries, which then means the determinant can only be 1 or -1. These are all the possible non-zero determinants and are not divisible by 3.