linear algebra - Problem on row Echelon form

Consider a $3\times 3$ matrix $$A =\begin{bmatrix}1 & 2 & -1\\2&1&0\\ 3&0&1\\\end{bmatrix}.$$ I have to find nonsingular matrix $P$ such that $PA$ is in row reduced Echelon form.

I am not able to get any idea to solve this problem. I understand Echelon form of a matrix. But what exactly should I do to solve this problem?

Thanks

Answer

We can obtain P in this way by left multiplication

$$P(I|A)=(P|PA)$$

thus consider $(I|A)$

$$\left[\begin{array}{rrr|rrr} 1 & 0 & 0 & 1 &2 & -1\\ 0 & 1 & 0 & 2 &1 &0 \\ 0 & 0 & 1 & 3 &0 &1\end{array}\right]$$

and by row operations

$$\left[\begin{array}{rrr|rrr} 1 & 0 & 0 & 1 &2 & -1\\ -2 & 1 & 0 & 0 &-3 &2 \\ -3 & 0 & 1 & 0 &-6 &4\end{array}\right]$$

$$\left[\begin{array}{rrr|rrr} 1 & 0 & 0 & 1 &2 & -1\\ -2 & 1 & 0 & 0 &-3 &2 \\ 1 & -2 & 1 & 0 &0 &0\end{array}\right]$$

thus

$$P =\begin{bmatrix}1 & 0 & 0\\-2&1&0\\ 1&-2&1\\\end{bmatrix}$$