Consider two matrices A = [[x1, x2, x3], [0, x2, x1], [0, 0, x3]] (upper triangular) and B = [[y1, 0, 0], [y3, y2, 0], [y2, y1, y3]] (lower triangular), where each xi, yi belongs to {-1, 0, 1}. If N is the number of ordered pairs of matrices (A, B) such that det(A) = det(B), find N/131.

Question

Accepted Answer

3. For upper triangular A: det(A) = x1*x2*x3. For lower triangular B: det(B) = y1*y2*y3. Each xi, yi in {-1,0,1}. For a triple from {-1,0,1}³: product = 0 if any entry is 0 (19 such triples out of 27), product = +1 for 4 triples, product = -1 for 4 triples. N = 19² + 4² + 4² = 361 + 16 + 16 = 393. N/131 = 3.

Consider two matrices A = [[x1, x2, x3], [0, x2, x1], [0, 0, x3]] (upper triangular) and B = [[y1, 0, 0], [y3, y2, 0], [y2, y1, y3]] (lower triangular), where each xi, yi belongs to {-1, 0, 1}. If N is the number of ordered pairs of matrices (A, B) such that det(A) = det(B), find N/131.

Solution

Related JEE Advanced Maths questions