Find the inverse of the matrix A = [[1,-1,0],[-1,0,1],[0,1,-1]].

GATE 1994 · Engineering Mathematics · Matrix · easy

Answer: The inverse does not exist because det(A) = 0 (matrix is singular; rows are linearly dependent: R1+R2+R3 = 0).

1. Check for row dependence: R1 + R2 + R3 = [1-1+0, -1+0+1, 0+1-1] = [0, 0, 0]. The three rows sum to zero, so they are linearly dependent.
2. Verify det(A) = 0: det(A) = 1*(0*(-1)-1*1) - (-1)*((-1)*(-1)-1*0) + 0*((-1)*1-0*0) = 1*(-1) - (-1)*(1) + 0 = -1 + 1 = 0.
3. Conclusion: Since det(A) = 0, A^(-1) does not exist. The matrix A is singular (non-invertible).