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Find two non-singular matrices P & Q such that PAQ is in normal form where$ A= \left[ \begin{array}{cccc} 1 & 2 & 3 & -4 \\ 2 & 1 & 4 & -5 \\ -1 & -5 & -5 & 7 \end{array}\right]$ Also find rank of A
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We write given matrix A as follows:

$ \left[ \begin{array}{cccc} 1 & 2 & 3 & -4 \\ 2 & 1 & 4 & -5 \\ -1 & -5 & -5 & 7 \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] \\ \; \\ \; \\ \; \\ R_2 \rightarrow R_2 - R_1 \; , \; R_3 \rightarrow R_3 + R_1 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 2 & 3 & -4 \\ 1 & -1 & 1 & -1 \\ 0 & -3 & -2 & 3 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 1 & 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_3 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & -1 & 1 & -1 \\ 1 & -1 & 1 & -1 \\ 0 & -3 & -2 & 3 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 2 & 0 & 1 \\ -1 & 1 & 0 \\ 1 & 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_2 \rightarrow R_2 - R_1 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & -1 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & -3 & -2 & 3 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 2 & 0 & 1 \\ -3 & 1 & -1 \\ 1 & 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_2 \longleftrightarrow R_3 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & -1 & 1 & -1 \\ 0 & -3 & -2 & 3 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 2 & 0 & 1 \\ 1 & 0 & 1 \\ -3 & 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_3 \rightarrow C_3-C_1 \; , \; C_4 \rightarrow C_4+C_1 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -3 & -2 & 3 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 2 & 0 & 1 \\ 1 & 0 & 1 \\ -3 & 1 & -1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & -1 & 1\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ C_4 \rightarrow C_4+C_2 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -3 & -2 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 2 & 0 & 1 \\ 1 & 0 & 1 \\ -3 & 1 & -1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 1 & -1 & 2\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ C_2 \rightarrow C_2-2C_3 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 2 & 0 & 1 \\ 1 & 0 & 1 \\ -3 & 1 & -1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 3 & -1 & 2\\ 0 & 1 & 0 & 0 \\ 0 & -2 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ C_3 \rightarrow C_3+C_3 \\ \; \\ \; \\ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \; = \; \left[ \begin{array}{ccc} 2 & 0 & 1 \\ 1 & 0 & 1 \\ -3 & 1 & -1 \end{array} \right] A \left[ \begin{array}{cccc} 1 & 3 & 5 & 2\\ 0 & 1 & 2 & 0 \\ 0 & -2 & -3 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ \therefore P \; = \; \left[ \begin{array}{ccc} 2 & 0 & 1 \\ 1 & 0 & 1 \\ -3 & 1 & -1 \end{array} \right] \\ \; \\ \; \\ Q \; = \; \left[ \begin{array}{cccc} 1 & 3 & 5 & 2\\ 0 & 1 & 2 & 0 \\ 0 & -2 & -3 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] \\ \; \\ \; \\ Rank \; \; of \; \; A \; = \;2 $

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