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 = bc)There is an x≠0 such that Fx = 0d)F must have two identical rowsCorrect answer is option 'D'. Can you explain this answer?

Question

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 &ne; 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 = bc)There is an x&ne;0 such that Fx = 0d)F must have two identical rowsCorrect answer is option 'D'. Can you explain this answer?

Answer

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.

Answer

(A) : Correct. We are given

Since

so we have a non-zero solution

to homogeneous equation

Now any vector

is also a solution of

and so we have infinitely many solutions of

and so determinant of F is zero.
(B) : Correct. Consider a vector

So there are infinitely many vectors of the form

which are solutions to equation Fx = b.
(C) : Correct. In option (a), we proved that vector

(D) : False. This is not necessary.
So option (D) is the answer.

Answer

And v are both solutions to the equation Fx = b.

To show that u = v, we can subtract the two equations:

Fu - Fv = b - b
Fu - Fv = 0

Now, let's consider the difference vector w = u - v.

If we substitute u = v + w into the equation Fu - Fv = 0, we get:

F(v + w) - Fv = 0
Fv + Fw - Fv = 0
Fw = 0

Since Fw = 0, it means that w is a solution to the homogeneous equation Fx = 0.

Now, let's use the fact that there are two distinct solutions, u and v, to the equation Fx = b.

If we substitute u = v + w into the equation Fx = b, we get:

F(v + w) = b
Fv + Fw = b

Since Fu = b and Fv = b, it means that Fw = 0.

Therefore, w must be the zero vector because it is a solution to the homogeneous equation Fx = 0.

Since w = u - v, if w is the zero vector, it means that u = v.

Therefore, u and v must be the same vector, and we have shown that u = v.

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