Which of the following statements is/are TRUE? (R denotes the set of real numbers.)
A. There exist M in R^(3x3), p in R^3, and q in R^3 such that Mx = p has a unique solution and Mx = q has no solution.
B. There exist M in R^(3x3), p in R^3, and q in R^3 such that Mx = p has no solution and Mx = q has infinite solutions.
C. There exist M in R^(3x3), p in R^3, and q in R^3 such that Mx = p has a unique solution and Mx = q has infinite solutions.
D. There exist M in R^(3x3), p in R^3, and q in R^3 such that Mx = p has no solution and Mx = q has infinite solutions.
GATE 2024 · Engineering Mathematics · System of Equations · medium
Answer: B and D are TRUE. Statements A and C are FALSE because a unique solution for any right-hand side forces M to be invertible, which then requires unique solutions for all right-hand sides.
Unique solution requires M to be invertible: If Mx = p has a unique solution, then the system has no free variables and M must have rank 3 (full rank). A full-rank 3x3 matrix is invertible. For an invertible M, Mx = b has a unique solution x = M^(-1)b for every b in R^3. Therefore Mx = q also has a unique solution for any q, making it impossible to have infinite solutions or no solution for q with the same M. Statements A and C are FALSE.
Singular M allows both no solution and infinite solutions: Let M = [[1,0,0],[0,1,0],[0,0,0]]. The column space is the xy-plane (all vectors with third component 0). Choose p = [0,0,1]^T: Mx = p requires 0 = 1, so no solution exists. Choose q = [1,1,0]^T (in the column space): Mx = q has solutions [1,1,t]^T for any t in R, giving infinitely many solutions. This example proves statements B and D are TRUE.