Let A^T denote the transpose of the matrix A = [[0, 0, a], [0, b, c], [d, e, f]], where a, b, c, d, e, f are integers with a*b*d not equal to 0. Find the number of such matrices for which A^(-1) = A^T.

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2³. A*A^T = I means rows of A are mutually orthogonal unit vectors with integer entries. Integer entries on unit sphere: each row must be a standard basis vector with entry +/-1 and rest 0. Given the structure of A with zeros at (1,1), (1,2) positions, a must be +/-1 (row 1: [0,0,a] is unit => a² = 1 => a = +/-1). Row 2: [0,b,c] unit => b² + c² = 1 with b != 0 => b = +/-1, c = 0. Row 3: [d,e,f] unit => d² + e² + f² = 1 with d != 0 => d = +/-1, e = 0, f = 0. Orthogonality of rows 1 and 2: a*c = 0 (satisfied since c=0). Rows 1 and 3: a*f = 0 (satisfied since f=0). Rows 2 and 3: b*e + c*f = 0 (satisfied since e=0, f=0). So a = +/-1 (2 choices), b = +/-1 (2 choices), d = +/-1 (2 choices). Total

Let A^T denote the transpose of the matrix A = [[0, 0, a], [0, b, c], [d, e, f]], where a, b, c, d, e, f are integers with abd not equal to 0. Find the number of such matrices for which A^(-1) = A^T.

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