(c) $\left[ {\begin{array}{*{20}{c}}4&1&0&0\\3&0&1&0\\0&0&0&0\end{array}} \right]$, (By ${R_3} \to {R_3} - 2{R_2})$
$ = \left[ {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&0\end{array}} \right],\,\,({\rm{By}}\,\,{C_1} \to {C_1} - 4{C_2} - 3{C_3})$
$ = \left[ {\begin{array}{*{20}{c}}1&0&0&0\\0&1&0&0\\0&0&0&0\end{array}} \right]$,
(Replace ${C_1}$ by ${C_2}$ and then Replace ${C_2}$ by ${C_3}$)
Hence rank of matrix is $2$