LA2.2.6

$A=\begin{bmatrix} 1 & 5 & 4 \\ 0 & 3 & 2 \\2 &13 & 10\end{bmatrix}\,$

$R_3-2R_1=\begin{bmatrix} 1 & 5 & 4 \\ 0 & 3 & 2 \\ 0 & 3 & 2\end{bmatrix}\,$

$R_2-R_3=\begin{bmatrix} 1 & 5 & 4 \\ 0 & 3 & 2 \\ 0 & 0 & 0\end{bmatrix}\,$

Rank of A is 2 since the number of non-zero rows is 2.

Now, $B=\begin{bmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\3 & 3 & 3\end{bmatrix}\,$

$R_2-2R_1, R_3 -3R_1 =\begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\,$

Therefore,Rank of B is 1.