The determinant of the matrix [ 2 0 0 0 ] [ 8 1 7 2 ] [ 2 0 2 0 ] [ 9 0 6 1 ] A. 4 B. 0 C. 15 D. 20

GATE 2000 · Engineering Mathematics · Determinant · easy

Answer: det = 4, so the answer is A. 4

1. Expand along column 2: Column 2 = [0, 1, 0, 0]. Only a_{22} = 1 is nonzero. So det(A) = 1 * (-1)^(2+2) * M_{22} = M_{22}.
2. Compute the 3x3 minor M_{22}: Remove row 2 and column 2 from the original matrix: rows 1,3,4 and columns 1,3,4 give [[2,0,0],[2,2,0],[9,6,1]]. This is lower triangular with diagonal entries 2, 2, 1.
3. Apply triangular determinant formula: Product of diagonal = 2 * 2 * 1 = 4. Therefore det(A) = 4.