Answer
a. False
b. False
c. True
d. True
e. True
Work Step by Step
a. False, the reduced echelon form is unique, but there can be many echelon forms. Here is a trivial counterexample, showing two row echelon forms of the same matrix:
$$
\begin{bmatrix}
1 & 1 & 4\\
0 & 1 & 3
\end{bmatrix}
\begin{bmatrix}
1 & 2 & 7 \\
0 & 1 & 3
\end{bmatrix}
$$
b. False, the reduced row echelon form is unique and thus so are the pivot positions.
c. True, by the definition of the forward phase of row reduction.
d. True, because if there are free variables, there can be infinite solutions (assuming the domain of is $\mathbb{R}$).
e. True, by definition.
