Question

Find the rank of the following matrix:
(a)$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 7\\ 3& 6& 10\end{array}\right]$.
(b)$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 4& 10& 18\end{array}\right]$.

Answer 1

A is 3 $\times$ 3 matrix and we can have minors of order 1, 2, 3.
Minor of order 3.
$\left|\begin{array}{ccc}1& 2& 3\\ 2& 4& 7\\ 3& 6& 10\end{array}\right|$
Apply $R_3-(R_1+R_2)$,
=$\left|\begin{array}{ccc}1& 2& 3\\ 2& 4& 7\\ 0& 0& 0\end{array}\right|$= 0
Minor of order 2.
$\left|\begin{array}{cc}1& 2\\ 2& 4\end{array}\right|$ = 0 $\left|\begin{array}{cc}2& 4\\ 3& 6\end{array}\right|$ = 0
$\left|\begin{array}{cc}2& 3\\ 4& 7\end{array}\right|$ = 14-12 = 2$\ne$0
Hence, there is at least one minor of order 2 which is not zero whereas all; the minors of order 2 + 1 i.e. 3 are zero. Hence p [A] = $2$.
(b) In this case minors of order $3=-2$ $\ne$ 0