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

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

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

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

Consider the following system of linear equations $$$\left[ {\matrix{ 2 & 1 & { - 4} \cr 4 & 3 & { - 12} \cr 1 & 2 & { - 8} \cr } } \right]\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ \alpha \cr 5 \cr 7 \cr } } \right]$$$

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of $$\alpha $$, does this system of equations have infinitely many solutions?