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$F$ is an $n\times n$ real matrix. $b$ is an $n\times 1$ real vector. Suppose there are two $n\times 1$ vectors, $u$ and $v$ such that, $u ≠ v$ and $Fu = b, Fv = b$. Which one of the following statements is false?

1.    Determinant of $F$ is zero.
2.    There are an infinite number of solutions to $Fx = b$
3.    There is an $x≠0$ such that $Fx = 0$
4.    $F$ must have two identical rows
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Ans is D. I am getting option A, B  be true and D  be false. I'm not understanding option C. Pls explain it

(A) : Correct. We are given

$$Fu = b$$

$$Fv = b$$

So $F(u-v) = 0$

Since $u \neq v$, so we have a non-zero solution $w = (u-v)$ to homogeneous equation $Fx=0$. Now any vector $\lambda w$ is also a solution of $Fx=0$, and so we have infinitely many solutions of $Fx=0$, and so determinant of F is zero.

(B) : Correct. Consider a vector $u+\lambda w$.

$$F(u+\lambda w) = Fu+F(\lambda w) = b + 0 = b$$

So there are infinitely many vectors of the form $u+\lambda w$, which are solutions to equation $Fx=b$.

(C) : Correct. In option (a), we proved that vector $(u-v) \neq 0$ satisfies equation $Fx=0$.

(D) : False. This is not necessary.

So option (D) is the answer.

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Having two identical rows means, that F determinant must be 0 which is same as (A). Then why (D) is considered as false?
Your statement is true, but its converse is not true i.e. if F determinant is 0, then it is not necessary that there must be two identical rows.
$u ≠ v$ and $Fu = b, Fv = b$ this tells that $Fx=b$  has infinite solutions, (either system of equation has unique solution or infinite solutions)

this is possible only when $\text{rank} \lt n$ that means determinant of $F$ is zero.

if $\text{rank} \lt n$ then of course $Fx=0$ also has infinite solutions.
How you assumed it's homogeneous equation?
there are more than two vectors satisfying equation FX=0 that means infinite solution exist...

determinant is 0 rank is less than n

but for determinant to be zero its not necessary to have two identical rows

option c is wrong?
here equation has infinite solutions so there exists vectors like y,z...etc others than given u v which will be satisfy Fx=0

"A matrix with two identical rows has a determinant of zero."
https://en.wikibooks.org/wiki/Linear_Algebra/Properties_of_Determinants

can someone explain option B and C in a simpler way