The sum of the elements in each row of matrix M is 1. If b = Mx - x for some vector x (with b != 0), which one of the following statements is correct?
A. The equation Mx = b has no solution
B. The equation Mx = b has exactly two solutions
C. The equation Mx = b has infinitely many solutions
D. The equation Mx = b has a unique solution
GATE 2025 · Engineering Mathematics · System of Equations · medium
Answer: C. The equation Mx = b has infinitely many solutions.
Eigenvalue 1 from row-stochastic property: Each row of M sums to 1, so M * e = e. This means e is an eigenvector of M with eigenvalue 1. Therefore 1 is an eigenvalue of M, which implies det(M - I) = 0, making M - I a singular matrix with a nontrivial null space.
b lies in range(M - I), so the system is consistent: Given b = Mx - x = (M - I)x, b is the image of the specific vector x under M - I. Thus x is already a particular solution to (M - I)y = b, confirming the system Mx = b is consistent (has at least one solution).
Consistent + singular => infinitely many solutions: Since M - I is singular, its null space is nontrivial (contains at least e, the all-ones vector). For any scalar t, the vector x + t*e is also a solution to (M - I)y = b: (M - I)(x + te) = (M - I)x + t*(M - I)e = b + t*0 = b. There are infinitely many such solutions, one for each value of t.