# Rank of a matrix

by proxyIP
Tags: matrix, rank
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,304 The simplest way to determine the rank of a matrix is to "row-reduce". The rank is the number of rows that contain non-zero entries. In the case you give $$A= \left(\begin{array}{ccc}1 & 2 & a \\-2 & 4a & 2\\ a & -2 & 1\end{array}\right)$$ Add twice the first row to the second and subtract a times the first row from the third to get $$A= \left(\begin{array}{ccc}1 & 2 & a \\0 & 4a+ 4 & 2+ 2a\\ 0 & -2-2a & 1-a^2\end{array}\right)$$ Now add half the second row to the third to get $$A= \left(\begin{array}{ccc}1 & 2 & a \\0 & 4a+ 4 & 2+ 2a\\ 0 & 0 & 2+ 2a- a^2\end{array}\right)$$ If a= -1 that has only 2 non-zero rows and so the rank of A is 2. If a= [itex]-1\pm\sqrt{3} the last row is 0 and again the rank of A is 2. For any other value of a, the rank is 3.