We write given matrix A as follows:-

$ \left[ \begin{array}{cccc} 1 & 2 & 3 & 2 \\ 2 & 3 & 5 & 1 \\ 1 & 3 & 4 & 5 \end{array} \right] \; = \; I_3AI_4 \; = \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 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] \\ \; \\ \; \\ C_3 \Rightarrow C_3-C_2 \\ \; \\ \left[ \begin{array}{cccc} 1 & 2 & 1 & 2 \\ 2 & 3 & 2 & 1 \\ 1 & 3 & 1 & 5 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ C_3 \Rightarrow C_3-C_1 \\ \; \\ \left[ \begin{array}{cccc} 1 & 2 & 0 & 2 \\ 2 & 3 & 0 & 1 \\ 1 & 3 & 0 & 5 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & -1 & 0\\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ R_2 \Rightarrow R_2-2R_1 \; , \; R_3 \Rightarrow R_3-R_1 \\ \; \\ \left[ \begin{array}{cccc} 1 & 2 & 0 & 2 \\ 0 & -1 & 0 & -3 \\ 0 & 1 & 0 & 3 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ -2 & 1 & 0 \\ -1& 0 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & -1 & 0\\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ R_3 \Rightarrow R_3+R_2 \\ \; \\ \left[ \begin{array}{cccc} 1 & 2 & 0 & 2 \\ 0 & -1 & 0 & -3 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ -2 & 1 & 0 \\ -3& 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & -1 & 0\\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ R_2 \Rightarrow R_2 \times (-1) \\ \; \\ \left[ \begin{array}{cccc} 1 & 2 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 2 & -1 & 0 \\ -3& 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & -1 & 0\\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ R_1 \Rightarrow R_1 - 2R_2 \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & -4 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccc} -3 & 2 & 0 \\ 2 & -1 & 0 \\ -3& 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & -1 & 0\\ 0 & 1 & -1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ C_4 \Rightarrow C_4 - 4C_1 \; , \; C_4 \Rightarrow C_4 - 3C_2 \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccc} -3 & 2 & 0 \\ 2 & -1 & 0 \\ -3& 1 & 1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 0 & -1 & 4\\ 0 & 1 & -1 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ \therefore P \; = \; \left[ \begin{array}{ccc} -3 & 2 & 0 \\ 2 & -1 & 0 \\ -3& 1 & 1 \end{array} \right] \\ \; \\ Q \; = \; \left[ \begin{array}{cccc} 1 & 0 & -1 & 4\\ 0 & 1 & -1 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ Rank \; of \; matrix \; A \; is \; 2. $