Let for $$A = \left[ {\matrix{ 1 & 2 & 3 \cr \alpha & 3 & 1 \cr 1 & 1 & 2 \cr } } \right],|A| = 2$$. If $$\mathrm{|2\,adj\,(2\,adj\,(2A))| = {32^n}}$$, then $$3n + \alpha $$ is equal to

If the system of equations

$$2 x+y-z=5$$

$$2 x-5 y+\lambda z=\mu$$

$$x+2 y-5 z=7$$

has infinitely many solutions, then $$(\lambda+\mu)^{2}+(\lambda-\mu)^{2}$$ is equal to

For the system of linear equations

$$2 x+4 y+2 a z=b$$

$$x+2 y+3 z=4$$

$$2 x-5 y+2 z=8$$

which of the following is NOT correct?

Let $$B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha > 2$$ be the adjoint of a matrix $$A$$ and $$|A|=2$$. Then $$\left[\begin{array}{ccc}\alpha & -2 \alpha & \alpha\end{array}\right] B\left[\begin{array}{c}\alpha \\ -2 \alpha \\ \alpha\end{array}\right]$$ is equal to :