Let $$A$$ be and n$$ \times $$n matrix of the folowing form.

What is the value of the determinant of $$A$$?

If matrix $$X = \left[ {\matrix{ a & 1 \cr { - {a^2} + a - 1} & {1 - a} \cr } } \right]$$
and $${X^2} - X + 1 = 0$$
($${\rm I}$$ is the identity matrix and $$O$$ is the zero matrix), then the inverse of $$X$$ is

In an M$$ \times $$N matrix such that all non-zero entries are covered in $$a$$ rows and $$b$$ columns. Then the maximum number of non-zero entries, such that no two are on the same row or column, is

The minimum number of colours required to colour the following graph, such that no two adjacent vertices are assigned the same colour, is