We write given matrix, say A such that $A=I_5AI_4$ , i.e.

$ \left[ \begin{array}{cccc} 2 & -1 & 1 & 1\\ 1 & 0 & 1 & 2 \\ 3 & 3 & 3 & 1 \\ 1 & 4 & 2 & 0 \\ 0 & -4 & -1 & -2 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ R_1 \rightarrow R_1-R_2 \; , \; R_3 \rightarrow R_3-3R_2 \; , \; R_4 \rightarrow R_4-R_2 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & -1 & 0 & -1\\ 1 & 0 & 1 & 2 \\ 0 & 3 & 0 & -5 \\ 0 & 4 & 1 & -2 \\ 0 & -4 & -1 & -2 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 \\ 0 & -3 & 1 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & -1 & 0 & -1\\ 1 & 0 & 1 & 2 \\ 0 & 3 & 0 & -5 \\ 0 & 4 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 \\ 0 & -3 & 1 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ R_2 \Rightarrow R_2-R_1 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & -1 & 0 & -1\\ 0 & 1 & 1 & 3 \\ 0 & 3 & 0 & -5 \\ 0 & 4 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ 0 & -3 & 1 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ C_2 \Rightarrow C_2+C_1 \; , \; C_4 \Rightarrow C_4+C_1 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 3 \\ 0 & 3 & 0 & -5 \\ 0 & 4 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ 0 & -3 & 1 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & 0 & 1\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ C_4 \Rightarrow C_4-3C_2 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 \\ 0 & 3 & 0 & -14 \\ 0 & 4 & 1 & -14 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ 0 & -3 & 1 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & 0 & -2\\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ R_3 \Rightarrow R_3-R_4 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & -1 & -1 & 0 \\ 0 & 4 & 1 & -14 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ 0 & -2 & 1 & -1 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & 0 & -2\\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ R_3 \Rightarrow R_2+R_3 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 4 & 1 & -14 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ -1 & 0 & 1 & -1 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & 0 & -2\\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ R_3 \Rightarrow R_2+R_3 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 4 & 1 & -14 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ -1 & 0 & 1 & -1 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & 0 & -2\\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ R_3 \longleftrightarrow R_4 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 4 & 1 & -14 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ -1 & 0 & 1 & -1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & 0 & -2\\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ C_3 \Rightarrow C_3-C_2 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 4 & -3 & -14 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 & 0 \\ -1 & 0 & 1 & -1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & -1 & -2\\ 0 & 1 & -1 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ R_3 \Rightarrow R_3-4R_2 \; , \; C_3 \Rightarrow C_3\times \dfrac{-1}{3} \; , \; C_4 \Rightarrow C_4\times \dfrac{-1}{14} \; \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ 4 & -9 & 0 & 1 & 0 \\ -1 & 0 & 1 & -1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & 1/3 & 2/14\\ 0 & 1 & 1/3 & 3/14 \\ 0 & 0 & -1/3 & 0 \\ 0 & 0 & 0 & -1/14 \end{array} \right] \\ \; \\ \; \\ \; \\ C_4 \Rightarrow C_4-C_3 \; \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ 4 & -9 & 0 & 1 & 0 \\ -1 & 0 & 1 & -1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & 1/3 & -4/21\\ 0 & 1 & 1/3 & -5/42 \\ 0 & 0 & -1/3 & 1/3 \\ 0 & 0 & 0 & -1/14 \end{array} \right] \\ \; \\ \; \\ \; \\ \; \\ \therefore P \; = \; \left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & 0\\ -1 & 2 & 0 & 0 & 0 \\ 4 & -9 & 0 & 1 & 0 \\ -1 & 0 & 1 & -1 & 0 \\ 0 & -1 & 0 & 1 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ \therefore Q \; = \; \left[ \begin{array}{cccc} 1 & 1 & 1/3 & -4/21\\ 0 & 1 & 1/3 & -5/42 \\ 0 & 0 & -1/3 & 1/3 \\ 0 & 0 & 0 & -1/14 \end{array} \right] \\ \; \\ \; \\ \; \\ Rank \; of \; matrix \; = \;3 $

Note: Values of P & Q may be different depending on transformations carried out. However, rank must be 3 in above example & PAQ must be equal to I.