Calculate Gauss-Jordan of this matrix Consider the matrix:
$$A=\begin{bmatrix} 0 & 1 &1 & 0 &-1 &0 \\ 0 & 3 & 3 &-2 &-3 &2 \\ 0 & -1&-1&1&1&-1 \end{bmatrix}\in M_{3\times6}(\mathbb Q)$$
(A) Calculate the normal Gauss-Jordan, $A '$, the matrix and calculate $P $ also invertible and such that $A' = P A$.
(B) Expressing rows of the matrix $A'$ as a combination of linear rows of $A$.
(C) Expressing rows of the matrix $A$ as a combination of linear rows of $A'$.
What is the meaning of "$\in M_{3\times6}(\mathbb Q) $"? I understand that is a matrix of $3\times6$.
Any help will be appreciated or any documentation in relation with the exercice.
 A: Perform the operation $-3R_1+R_2$ and $R_1+R_3$, I obtain the following matrices
I obtain the matrix $$\begin{bmatrix} 0 & 1 & 1 & 0 & -1 & 0 \\ 0 & 0  & 0 & -2 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 & -1 &\end{bmatrix}$$
Next, I perform the operation $2R_3+R_2,$
I obtain the matrix $$\begin{bmatrix} 0 & 1 & 1 & 0 & -1 & 0 \\ 0 & 0  & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & -1 &\end{bmatrix}$$
Last but not least, we perform the operation $R_2 \leftrightarrow R_3,$
I obtain the matrix $$A'=\begin{bmatrix} 0 & 1 & 1 & 0 & -1 & 0 \\ 0 & 0  & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 &\end{bmatrix}.$$
To recover the $P$, peform the same operation on matrix $I_3$
\begin{align}
I_3 &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \\
& \xrightarrow{-3R_1+R_2}\begin{bmatrix} 1 & 0 & 0 \\ -3 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \\
& \xrightarrow{R_1+R_3}\begin{bmatrix} 1 & 0 & 0 \\ -3 & 1 & 0 \\ 1 & 0 & 1\end{bmatrix} \\
& \xrightarrow{2R_3+R_2}\begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 2 \\ 1 & 0 & 1\end{bmatrix} \\
& \xrightarrow{R_2 \leftrightarrow R_3}\begin{bmatrix} 1 & 0 & 0  \\ 1 & 0 & 1 \\ -1 & 1 & 2\end{bmatrix}=P \\
\end{align}
Now that you have obtain the matrix $P$, you can express rows of $A'$ as linear combination of rows of $A$.
Also note that $A=P^{-1}A'$, that is if you can compute the inverse of $P$ (either by computing the inverse directly or reversing the elementaries operations), then you can express rows of $A$ as linear combination of rows of $A'$.
