Let A be a 3 $$\times$$ 3 real matrix such that

$$A\left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 0 \cr } } \right);A\left( {\matrix{ 1 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ { - 1} \cr 0 \cr 1 \cr } } \right)$$ and $$A\left( {\matrix{ 0 \cr 0 \cr 1 \cr } } \right) = \left( {\matrix{ 1 \cr 1 \cr 2 \cr } } \right)$$.

If $$X = {({x_1},{x_2},{x_3})^T}$$ and I is an identity matrix of order 3, then the system $$(A - 2I)X = \left( {\matrix{ 4 \cr 1 \cr 1 \cr } } \right)$$ has :

Let $$A = \left[ {\matrix{ 0 & { - 2} \cr 2 & 0 \cr } } \right]$$. If M and N are two matrices given by $$M = \sum\limits_{k = 1}^{10} {{A^{2k}}} $$ and $$N = \sum\limits_{k = 1}^{10} {{A^{2k - 1}}} $$ then MN² is :

Let the system of linear equations

x + y + $$\alpha$$z = 2

3x + y + z = 4

x + 2z = 1

have a unique solution (x$$^ * $$, y$$^ * $$, z$$^ * $$). If ($$\alpha$$, x$$^ * $$), (y$$^ * $$, $$\alpha$$) and (x$$^ * $$, $$-$$y$$^ * $$) are collinear points, then the sum of absolute values of all possible values of $$\alpha$$ is

The number of values of $$\alpha$$ for which the system of equations :

x + y + z = $$\alpha$$

$$\alpha$$x + 2$$\alpha$$y + 3z = $$-$$1

x + 3$$\alpha$$y + 5z = 4

is inconsistent, is