a
(a) Given $A = \left[ {\begin{array}{*{20}{c}}2&3&1&4\\0&1&2&{ - 1}\\0&{ - 2}&{ - 4}&2\end{array}} \right]$, $({R_2} \to 2{R_2} + {R_3})$
$A = \left[ {\begin{array}{*{20}{c}}2&3&1&4\\0&0&0&0\\0&{ - 2}&{ - 4}&2\end{array}} \right]$
Since every minor of order $3$ in $ A$ is $0$ and there exists a minor order $3 $
i.e. $\left[ {\begin{array}{*{20}{c}}2&3\\0&{ - 2}\end{array}} \right]$ in $ A$ which is non-zero.
Thus, rank = $2.$