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)$\left[\begin{array}{ccc}1& 3& 2\\ 1& 2& 3\\ 1& 5& 4\end{array}\right]$.
(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)$\left[\begin{array}{ccc}1& -3& 2\\ 3& -9& 6\\ -2& 6& -4\end{array}\right]$.

Answer 1

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