Let M(x) = bmatrix 1 & 2x & 2x^2 0 & 1 & 2x 0 & 0 & 1 bmatrix for any real number x . Let P = M(1) M(2) M(10) and Q = P⁻¹ . The sum of the elements in the first row of the matrix Q is:

Step-by-step solution

First, let us find the product of two matrices M(x) and M(y) : M(x)M(y) = bmatrix 1 & 2x & 2x^2 0 & 1 & 2x 0 & 0 & 1 bmatrix bmatrix 1 & 2y & 2y^2 0 & 1 & 2y 0 & 0 & 1 bmatrix Multiplying the matrices: Row 1, Column 1: 1 1 + 2x 0 + 2x^2 0 = 1 Row 1, Column 2: 1 2y + 2x 1 + 2x^2 0 = 2(x+y) Row 1, Column 3: 1 2y^2 + 2x 2y + 2x^2 1 = 2(x^2 + 2xy + y^2) = 2(x+y)^2 Row 2, Column 2: 1 Row 2, Column 3: 1 2y + 2x 1 = 2(x+y) Row 3, Column 3: 1 Thus, M(x)M(y) = bmatrix 1 & 2(x+y) & 2(x+y)^2 0 & 1 & 2(x+y) 0 & 0 & 1 bmatrix =