If the system of equations

$$ \begin{aligned} & (\lambda-1) x+(\lambda-4) y+\lambda z=5 \\ & \lambda x+(\lambda-1) y+(\lambda-4) z=7 \\ & (\lambda+1) x+(\lambda+2) y-(\lambda+2) z=9 \end{aligned}$$

has infinitely many solutions, then $\lambda^2+\lambda$ is equal to

If $\mathrm{A}, \mathrm{B}, \operatorname{and}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of $A\left(\operatorname{adj}\left(A^{-1}\right)+\operatorname{adj}\left(B^{-1}\right)\right)^{-1} B$, is equal to

If the system of linear equations :

$$\begin{aligned} & x+y+2 z=6 \\ & 2 x+3 y+\mathrm{az}=\mathrm{a}+1 \\ & -x-3 y+\mathrm{b} z=2 \mathrm{~b} \end{aligned}$$

where $a, b \in \mathbf{R}$, has infinitely many solutions, then $7 a+3 b$ is equal to :

For a $3 \times 3$ matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A|=\frac{1}{2}$ and trace $(A)=3$. If $B=\operatorname{adj}(\operatorname{adj}(2 A))$, then the value of $|B|+$ trace $(B)$ equals :