row-echelon form

[AerospaceEng]

quick question if i have the matrix; (R represents a specific row)

1 1 3 -3 0
0 2 1 -3 3
0 -1 -1 2 -1

R3+R2 -->R2

1 1 3 -3 0
0 1 0 -1 2
0 -1 -1 2 -1

and then R2+R3 -->R3

1 1 3 -3 0
0 1 0 -1 2
0 0 -1 1 1

is there anything wrong with my adding of rows or what not? I'm really just second guessing if i can add a lower row to a higher row. thanks.

[HallsofIvy]

I think it is far better to use a simple "algorithm" always doing things the same way, working on one column at a time, from left to rightm top to bottom.

That said, I would not Use your first step where you add the third row to the second. Instead I would either "divide the second row by 2" then "add the new second row to the third" which gives
\begin{bmatrix}1 & 1 & 3 & -3 & 0 \\
0 & 1 & 1/2 & -3/2 & 3/2 \\
0 & 0 & -1/2 & 1/2 & 1/2 \end{bmatrix}

or "swap the second and third rows" then "add twice the new second row to the new third row".

\begin{bmatrix}1 & 1 & 3 & -3 & 0 \\
0 & -1 & -1 & 2 & -1 \\
0 & 0 & -1 & 1 & 1\end{bmatrix}

[Bacle]

quick question if i have the matrix; (R represents a specific row)

1 1 3 -3 0
0 2 1 -3 3
0 -1 -1 2 -1

R3+R2 -->R2

1 1 3 -3 0
0 1 0 -1 2
0 -1 -1 2 -1

and then R2+R3 -->R3

1 1 3 -3 0
0 1 0 -1 2
0 0 -1 1 1

is there anything wrong with my adding of rows or what not? I'm really just second guessing if i can add a lower row to a higher row. thanks.

You do add multiple of lower rows to higher rows, after reducing, to turn your matrix
into reduced row-echelon.