Consider the system, each consisting of m linear equations in $$n$$ variables.
$$I.$$ $$\,\,\,$$ If $$m < n,$$ then all such system have a solution
$$II.$$ $$\,\,\,$$ If $$m > n,$$ then none of these systems has a solution
$$III.$$ $$\,\,\,$$ If $$m = n,$$ then there exists a system which has a solution

Which one of the following is CORRECT?

Suppose that the eigen values of matrix $$A$$ are $$1, 2, 4.$$ The determinant of $${\left( {{A^{ - 1}}} \right)^T}$$ is _______.

Two eigenvalues of a $$3 \times 3$$ real matrix $$P$$ are $$\left( {2 + \sqrt { - 1} } \right)$$ and $$3.$$ The determinant of $$P$$ is _______.

Let $${a_n}$$ be the number of $$n$$-bit strings that do NOT contain two consecutive $$1s.$$ Which one of the following is the recurrence relation for $${a_n}$$?