Question

Find the rank the following matrices:
(a) $A=\left[\begin{array}{ccc}1& 3& 4\\ 2& 6& 8\end{array}\right]$
(b) $A=\left[\begin{array}{ccc}3& -1& 2\\ -6& 2& -4\\ -3& 1& -2\end{array}\right]$

Answer 1

(a) $A$ is $2\times 3$ matrix and we can have minors of orders 1 and 2. Minor of order 2.
$\left|\begin{array}{cc}1& 3\\ 2& 6\end{array}\right|=6-6=0$
Other minors of order 2 are
$\left|\begin{array}{cc}3& 4\\ 6& 8\end{array}\right|=0$ and $\left|\begin{array}{cc}1& 4\\ 2& 8\end{array}\right|=0$
Thus all minors of order 2 zero, beacuse of identical lines.
Hence, we have only minors of order 1, which will be elements of the given matrix and are not all equal to zero or at least one of them is not equal to zero. Hence, the rank $A$ is 1.
(b) Proceed as above. The minors of order 3 and 2 all zero because of identical lines.
$R_2=-2R_1,R_3=-R_1$ $\therefore$ Rank $A$ = 1