What is the minimum number of multiplications involved in computing the matrix product $$PQR?$$ Matrix $$P$$ has $$4$$ rows and $$2$$ columns, matrix $$Q$$ has $$2$$ rows and $$4$$ columns and matrix $$R$$ has $$4$$ rows and $$1$$ column ______.

The inverse of the matrix $$\left[ {\matrix{ {3 + 2i} & i \cr { - i} & {3 - 2i} \cr } } \right]$$ is

The minimum and maximum eigen values of matrix $$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \cr } } \right]$$ are $$-2$$ and $$6$$ respectively. What is the other eigen value?

For what values of $$\alpha $$ and $$\beta $$ the following simultaneous equations have an infinite number of solutions $$$x+y+z=5,$$$ $$$x+3y+3z=9,$$$ $$$x + 2y + \alpha z = \beta $$$