The rank of the matrix left[ {begin{array}{*{ | vedclass

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(C) Let the given matrix be $A = \left[ {\begin{array}{*{20}{c}}4&1&0&0\3&0&1&0\6&0&2&0\end{array}} ight]$.Apply the row operation ${R_3} 	o {R_3

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$2$

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(C) Let the given matrix be $A = \left[ {\begin{array}{*{20}{c}}4&1&0&0\3&0&1&0\6&0&2&0\end{array}} ight]$.Apply the row operation ${R_3} 	o {R_3} - 2{R_2}$:$A \sim \left[ {\begin{array}{*{20}{c}}4&1&0&0\3&0&1&0\6-2(3)&0-2(0)&2-2(1)&0-2(0)\end{array}} ight] = \left[ {\begin{array}{*{20}{c}}4&1&0&0\3&0&1&0\0&0&0&0\end{array}} ight]$.Since there are two non-zero rows in the row-echelon form,the rank of the matrix is $2$.Alternatively,we can observe that the third row is $2$ times the second row $(R_3 = 2R_2)$,which means the rows are linearly dependent. The first two rows are linearly independent. Thus,the rank is $2$.

The rank of the matrix $\left[ {\begin{array}{*{20}{c}}4&1&0&0\\3&0&1&0\\6&0&2&0\end{array}} \right]$ is:

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