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 $${a_n}$$?

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 _______.

In the LU decomposition of the matrix $$\left[ {\matrix{ 2 & 2 \cr 4 & 9 \cr } } \right]$$, if the diagonal elements of U are both 1, then the lower diagonal entry $${l_{22}}$$ of L is ________.