Answer
See answer below
Work Step by Step
We can reduce A to row-echelon form using the following sequence of elementary row operations:
$\begin{bmatrix}
3& -1 & 4\\
2& 1 & 3\\
1 & 3 & 2
\end{bmatrix} \approx^1\begin{bmatrix}
1 & 3 & 2\\
2& 1 & 3\\
3& -1 & 4
\end{bmatrix} \approx^2\begin{bmatrix}
1 & 3 & 2\\
0& -5 & -1\\
3& -1 & 4
\end{bmatrix} \approx^3 \begin{bmatrix}
1 & 3 & 2\\
0& -5 & -1\\
0& -10 & -2
\end{bmatrix} \approx^4 \begin{bmatrix}
1 & 3 & 2\\
0& 1 & \frac{1}{5}\\
0& -10 & -2
\end{bmatrix} \approx^5 \begin{bmatrix}
1 & 3 & 2\\
0& 1 & \frac{1}{5}\\
0& 0 & 0
\end{bmatrix}$
$1. P_{13}$
$2.A_{12}(-2)$
$3.A_{13}(-3)$
$4.M_2( \frac{-1}{5})$
$5.A_{23}(10)$
So, elementary matrices are $A_{23}(10),M_2( \frac{-1}{5}),A_{13}(-3),A_{12}(-2),P_{13}=$
