Then det(C) will be:

To find the determinant of matrix C, observe the relationship between its columns. Let C1, C2, and C3 be the first, second, and third columns respectively. Calculate the sum of Column 1 and Column 3: C1 + C3 = (sin^2θ, cos^2θ, -3)^T + (cos^2θ, sin^2θ, 7)^T = (sin^2θ + cos^2θ, cos^2θ + sin^2θ, -3 + 7)^T. Using the identity sin^2θ + cos^2θ = 1, this simplifies to (1, 1, 4)^T. This result is identical to Column 2 (C2). Since one column is the sum of the other two, the columns are linearly dependent. Therefore, det(C) = 0.