The Rank of the matrix is equal to the number of linear independent columns in the matrix.
We know that , the pivot column (leading entry of non zero row) is linearly independent column.
So we will convert this matrix into Echelon form using Gauss elimination method.
$\begin{bmatrix} 1 & 2& 3& 4\\ 5 & 6& 7& 8\\ 6& 8& 10&12 \\ 151& 262& 373&484 \end{bmatrix}$
Steps:-
1.We will make all the entries in column below a leading entry(here it is 1) ,zero
R2 $\leftarrow$ R2 – 5R1
R3 $\leftarrow$ R3 – 6R1
R4 $\leftarrow$ R4 – 151R1
$\begin{bmatrix} 1 &2 &3 &4 \\ 0&-4 &-8 &-12 \\ 0 & -4 & -8 &-12 \\ 0 & 0& 0& 0 \end{bmatrix}$
- Now make all the entries in column below a leading entry (here it is -4) zero
R3$\leftarrow$ R3 – R2
$\begin{bmatrix} 1 &2 &3 &4 \\ 0 &-4 &-8 &-12 \\ 0& 0 & 0 &0 \\ 0& 0 &0 &0 \end{bmatrix}$
Now, since there are two Pivot columns (i.e C1 and C2)
therefore there are two linear independent coloumn which means the Rank is 2
$\therefore$ B) 2