If I tell you that each of the four numbers 1242,1426,2070,and 1564
is divisible by 23(trust me!), could you with no explicit calculation prove that the determinant of the 4x4 matrix
1 2 4 2
1 4 2 6
1 5 6 4
2 0 7 0
is also divisible by 23?

The determinant of 4x4 matrix:
1 2 4 2
1 4 2 6
1 5 6 4
2 0 7 0
equals the determinant of
1 2 4 1242
1 4 2 1426
1 5 6 1564
2 0 7 2070,
Since all members of the last column are divisible by 24, so is the determinant. q.e.d.

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