The product of a 9 × 4 matrix and a 4 × 9 matrix contains a variable x in exactly two places. If D x is the determinant of the matrix product such that D 0 = 1 , D - 1 = 1 and D 2 = 7 , then D - 2 is equal to

Question

The product of a 9 × 4 matrix and a 4 × 9 matrix contains a variable x in exactly two places. If D x is the determinant of the matrix product such that D 0 = 1 , D - 1 = 1 and D 2 = 7 , then D - 2 is equal to,

MARKS App · Accepted Answer

Let, the order of matrix A is 9 × 4 and matrix B is 4 × 9 D x = A B = a polynomial of degree 2 as it has x at 2 places. Let, D x = a x 2 + b x + c D 0 = c = 1 D - 1 = a - b + c = 1 ⇒ a - b = 0 D 2 = 4 a + 2 b + c = 7 ⇒ 4 a + 2 b = 6 ⇒ 2 a + b = 3 Solving these two equations, we get, a = 1 , b = 1 ⇒ D x = x 2 + x + 1 ⇒ D - 2 = 4 - 2 + 1 = 3

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