I have the following question here:
A certain linear system $Ax=b$ consisting of $n$ unknowns has two distinct solutions $x_1$ and $x_2$ with $x_1 \neq x_2$. Which is the following statements is false?
$(A)$ The reduced row echelon form of $A$ is the $n \times n$ identity matrix.
$(B)$ The reduced row echelon form of $A$ has a row of zeros.
$(D)$ There are an infinite number of distinct solutions of this system.
$(E)$ A is not invertible.
The correct answer is $(A)$ however I am not sure why. How are they justifying that this is the identity matrix after row reduction?
In any case, how are the other statements true?
It would be much appreciated if someone could explain each answer choice to me. Thanks!