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Reduce matrix A to normal form and find its rank where $\\ \; \\ A=\left[ \begin{array}{cccc} 1 & 2 & 3 & 2 \\ 2 & 3 & 5 & 1 \\ 1 & 3 & 4 & 5 \end{array} \right]$

Mumbai University > First Year Engineering > Sem 1 > Applied Maths 1

Marks : 6 M

Year : May 2013

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A = $\left[ \begin{array}{ccc} 1& 2& 3& 2\\ 2& 3& 5& 1\\ 1& 3& 4& 5 \end{array}\right] \\ \; \\ \; \\ \; \\ R_2 \rightarrow R_2-2R_1 \\ R_3 \rightarrow R_3-R_1 \\ \; \\ \; \\ A\;=\; \left[ \begin{array}{ccc} 1& 2& 3& 2\\ 0& -1& -1& -3\\ 0& 1& 1& 3 \end{array}\right] \\ \; \\ \; \\ \; \\ R_3 \rightarrow R_3+R_2 \\ R_1 \rightarrow R_1+2R_2 \\ \; \\ \; \\ A\;=\; \left[ \begin{array}{ccc} 1& 0& 1& -4\\ 0& -1& -1& -3\\ 0& 0& 0& 0 \end{array}\right] \\ \; \\ \; \\ \; \\ C_3 \rightarrow C_3-C_1 \\ C_4 \rightarrow C_4+4C_1 \\ \; \\ \; \\ A\;=\; \left[ \begin{array}{ccc} 1& 0& 0& 0\\ 0& -1& -1& -3\\ 0& 0& 0& 0 \end{array}\right] \\ \; \\ \; \\ \; \\ C_2 \rightarrow C_2-C_3 \\ C_4 \rightarrow C_4-3C_3 \\ \; \\ \; \\ A\;=\; \left[ \begin{array}{ccc} 1& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 0 \end{array}\right] \\ \; \\ \; \\ \; \\ C_3 \rightarrow C_2/-1 \\ \; \\ \; \\ A\;=\; \left[ \begin{array}{ccc} 1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0 \end{array}\right] \\ \; \\ \; \\ \;$

Thus matrix A has been reduced to normal form

The number of non-zero rows in normal form of matrix A is 2

Therefore, the rank of the matrix is 2

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