Question

Find the rank of the following matrices:
(a) $\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 8\\ 4& 10& 18\end{array}\right]$
(b) $\left[\begin{array}{ccc}1& 3& 2\\ 1& 3& 2\\ 1& 5& 4\end{array}\right]$
(c) $\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 4& 5& 6\end{array}\right]$
(d) $\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ -2& -3& -1\end{array}\right]$
(e) $\left[\begin{array}{ccc}13& 16& 19\\ 14& 17& 20\\ 15& 18& 21\end{array}\right]$
(f) $\left[\begin{array}{ccc}1& -3& 3\\ 3& -9& 6\\ -2& 6& -4\end{array}\right]$

Answer 1

(a) $|A|=2\ne 0$ $\therefore \rho \left(A\right)=3$
(b) Apply $R_2-R_1,R_3-R_1$ $\therefore |A|=-4\ne 0$
$\therefore \rho \left(A\right)=3$
(c) $R_3-R_2,R_2-R_1$ makes two rows identical.
$\therefore |A|=0$
minor of order 2 is not zero
$\rho \left(A\right)=2$
(d) $R_2+R_3$ gives $|A|=0$
$\therefore \rho \left(A\right)=2$ as above
(e) $\rho \left(A\right)=2$ exactly as in part (c)
(f) $R_3+2R_1$ amkes $R_3$ a row of zeros.
$\therefore |A|=0$
$\therefore \rho \left(A\right)=2$ as above.