$$ \text { The rank of } A=\left[\begin{array}{ccc} 1 & x & x+1 \\ 2 x & x^2-x & x^2+x \\ 3 x(x-1) & x\left(x^2-3 x+2\right) & x\left(x^2-1\right) \end{array}\right] \text { is } $$

Let $I$ be a unit matrix of order 6 . Let $A=\left(a_{i j}\right)$ be a square matrix of order 6 such that $a_{i j}=\left\{\begin{array}{l}1, \text { if } i+j=7 \\ 0, \text { if } i+j \neq 7\end{array}\right.$ then $\left(A(\operatorname{adj} A) A^{-1}\right) A^2=$

Let $a, b, c \notin\{0,1\}$. If the system of equations

$$ \begin{aligned} & \Pi_1 \equiv x+a y+a z=0 \\ & \Pi_2 \equiv b x+y+b z=0 \\ & \Pi_3 \equiv c x+c y+z=0 \end{aligned} $$

has a non-trivial solution, then the system of equations $\Pi_1=a, \Pi_2=b, \Pi_3=c$ has

$A$ is a singular matrix of order five. $B$ is another matrix having the rank $\rho(B)$ equal to the $\operatorname{rank} \rho(A)$ and $B$ has a non-zero minor of order 3. Then which one of the following is true?