F is an n*n real matrix. b is an n*1 real vector. Suppose there are two n*1 vectors, u and v such that, u ≠ v and Fu = b, Fv = b. Which one of the following statements is false?
(A) Determinant of F is zero.
(B) There are an infinite number of solutions to Fx = b
(C) There is an x≠0 such that Fx = 0
(D) F must have two identical rows
Answer: (D)
Explanation: Since Fu = b, and also Fv = b, so we have (Fu – Fb) = 0 i.e. F(u-v) = 0. Since u≠v, F is a singular matrix i.e. its determinant is 0. Now for a singular matrix F, either Fx = b has no solution or infinitely many solutions, but as we are already given two solutions u and v for x, Fx = b has to have infinitely many solutions.
Moreover, by definition of singular matrix, there exists an x≠0 such that Fx = 0 .
So options (A), (B), and (C) are true. Option (D) is false because it may not be necessary that two rows are identical, instead, two columns can be identical and we can get F as singular matrix then.
So option (D) is correct answer.
Source: http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2006.html
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