The determinant of the matrix given below is [ 0 1 0 2 ] [-1 1 1 3 ] [ 0 0 0 1 ] [ 1 -2 0 1 ] A. -1 B. 0 C. 1 D. 2

GATE 2005 · Engineering Mathematics · Determinant · medium

Answer: -1

1. Expand along column 3: Column 3 entries: a_{13}=0, a_{23}=1, a_{33}=0, a_{43}=0. Only a_{23}=1 contributes. det(A) = 1 * (-1)^(2+3) * M_{23} = -M_{23}.
2. Form submatrix for M_{23}: Remove row 2 ([-1,1,1,3]) and column 3 (the 0,1,0,0 column). Remaining rows: row1=[0,1,2], row3=[0,0,1], row4=[1,-2,1]. So M_{23} = det([[0,1,2],[0,0,1],[1,-2,1]]).
3. Compute M_{23} by expanding along column 1: Column 1 entries: 0, 0, 1. Only row3 entry (1) contributes: M_{23} = 1 * (-1)^(3+1) * det([[1,2],[0,1]]) = 1 * 1 * (1*1 - 2*0) = 1.
4. Final determinant: det(A) = -1 * 1 = -1.