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  1. GATE CS
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  3. Engineering Mathematics

Consider the following system of linear equations: 2x_1 - x_2 + 3x_3 = 1 3x_1 + 2x_2 + 5x_3 = 2 -x_1 + 4x_2 + x_3 = 3 The system of equations has A. no solution B. a unique solution C. more than one but a finite number of solutions D. an infinite number of solutions

GATE 2005 · Engineering Mathematics · System of Equations · medium

Answer: det(A) = 14 != 0, so the system has a unique solution. Answer: B.

  1. Compute det(A) by cofactor expansion along row 1: A = [[2,-1,3],[3,2,5],[-1,4,1]]. det = 2*det([[2,5],[4,1]]) - (-1)*det([[3,5],[-1,1]]) + 3*det([[3,2],[-1,4]]). det([[2,5],[4,1]]) = 2*1 - 5*4 = 2 - 20 = -18. det([[3,5],[-1,1]]) = 3*1 - 5*(-1) = 3+5 = 8. det([[3,2],[-1,4]]) = 3*4 - 2*(-1) = 12+2 = 14.
  2. Assemble the determinant: det(A) = 2*(-18) + 1*(8) + 3*(14) = -36 + 8 + 42 = 14. Since det(A) = 14 != 0, the matrix is non-singular.
  3. State conclusion: Since det(A) != 0, rank(A) = 3 = n. Therefore rank(A) = rank([A|b]) = n, so the system has a unique solution for any b.