Answer
multiple answers possible,
please see step-by-step
Work Step by Step
Use matrix row operations to get ls down the main diagonal from upper left to lower right, and Os below the $1\mathrm{s}$.
Transition first to second matrix:
we want 0s below the 1 at $a_{11}$
The matrix row operations are found when you ask:
"How to 0 at $a_{21}$ and $a_{31}$?"
$-2R1+R2\rightarrow R2\quad $and $\quad 3R1+R3\rightarrow R3$
The second matrix is
$\left[\begin{array}{lllll}
1 & -2 & 3 & | & 4\\
0 & 5 & [-10] & | & [-5]\\
0 & -2 & [8] & | & [10]
\end{array}\right]$
Next step: we want 1 on the diagonal, at $a_{22}.$
(*) One way: $\displaystyle \frac{1}{5}R_{2}\rightarrow R2$
$\left[\begin{array}{lllll}
1 & -2 & 3 & | & 4\\
0 & 1 & [-2] & | & [-1]\\
0 & -2 & [8] & | & [10]
\end{array}\right] $
(offer this as answer, reading the missing elements in brackets)
(**) Another possible row operation: $2R3+R2\rightarrow R2$
(to place a 1 on the diagonal at $a_{22}):$
$\left[\begin{array}{lllll}
1 & -2 & 3 & | & 4\\
0 & 1 & [6] & | & [19]\\
0 & -2 & [8] & | & [10]
\end{array}\right]$
