[Math] For which values of $a, b, c, d,$ and $e$ is the following matrix in reduced row-echelon form

Question

What are the values of $a, b, c, d,$ and $e$ of the following matrix in reduced row echelon form? I guess $e$ should be $0$.

$$\left[\begin{array}{c}0 & a & 2 & 1 & b\\ 0 & 0 & 0 & c & d\\ 0 & 0 & e & 0 & 0\end{array}\right]\\$$

Answer 1

If it is reduced row echelon form then the first non-zero in each row must equal $1$ and all other entries in that column must be zero. Each "first $1$" in a row must be to the right of any first $1$ in rows above it. Any rows of all zeros must be at the bottom.

So $a=1$ and $c=0$. Either $d=0$ or $d=1$. If $d=0$ then $b$ may take any value, but of $d=1$ then $b=0$.

$e$ must equal $0$ as you suggested.

Thus the two correct solutions are

$\left(\begin{array}{c}0 & 1 & 2 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0\end{array}\right)\\\left(\begin{array}{c}0 & 1 & 2 & 1 & b\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\end{array}\right)\\$