Let $$A$$ be an $$m\,\, \times \,\,n$$ matrix and $$B$$ an $$n\,\, \times \,\,m$$ matrix. It is given that determinant $$\left( {{{\rm I}_m} + AB} \right) = $$determinant $$\left( {{{\rm I}_n} + BA} \right),$$ where $${{{\rm I}_k}}$$ is the $$k \times k$$ identity matrix. Using the above property, the determinant of the matrix given below is $$\left[ {\matrix{ 2 & 1 & 1 & 1 \cr 1 & 2 & 1 & 1 \cr 1 & 1 & 2 & 1 \cr 1 & 1 & 1 & 2 \cr } } \right]$$

The minimum eigenvalue of the following matrix is $$\left[ {\matrix{ 3 & 5 & 2 \cr 5 & {12} & 7 \cr 2 & 7 & 5 \cr } } \right]$$

Given that $$A = \left[ {\matrix{ { - 5} & { - 3} \cr 2 & 0 \cr } } \right]$$ and $${\rm I} = \left[ {\matrix{ 1 & 0 \cr 0 & 1 \cr } } \right],$$ the value of $${A^3}$$ is

The system of equations $$x+y+z=6,$$ $$x+4y+6z=20,$$ $$x + 4y + \lambda z = \mu $$ has no solution for values of $$\lambda $$ and $$\mu $$ given by