A115222 Number of 3 X 3 matrices over Z(n) having nonzero determinant.

0, 168, 11232, 161280, 1488000, 7170552, 33784128, 106659840, 327525120, 839997000, 2124276000, 4307378688, 9726417792, 18401146008, 34512588000

Offset: 1

Author

T. D. Noe, Jan 16 2006

Keywords

• nonn

Comments

See A064767 for the number of invertible matrices.

Programs

Mathematica

Table[cnt=0; Do[m={{a, b, c}, {d, e, f}, {g, h, i}}; If[Det[m, Modulus->n]>0, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}, {d, 0, n-1}, {e, 0, n-1}, {f, 0, n-1}, {g, 0, n-1}, {h, 0, n-1}, {i, 0, n-1}]; cnt, {n, 2, 7}]

Formula

a(n)=n^9-A115221(n). For prime n, a(n)=A064767(n)=n^3 (n-1)^3 (n+1) (n^2+n+1)