Consider the following $$2 \times 2$$ matrix $$A$$ where two elements are unknown and are marked by $$a$$ and $$b.$$ The eigenvalues of this matrix ar $$-1$$ and $$7.$$ What are the values of $$a$$ and $$b$$?
$$A = \left( {\matrix{ 1 & 4 \cr b & a \cr } } \right)$$

For a set A, the power set of A is denoted by 2^A. If A = {5, {6}, {7}}, which of the following options are TRUE?

I. $$\phi \in {2^A}$$
II. $$\phi \subseteq {2^A}$$
III. $$\left\{ {5,\left\{ 6 \right\}} \right\} \in {2^A}$$
IV. $$\left\{ {5,\left\{ 6 \right\}} \right\} \subseteq {2^A}$$

Suppose L = { p, q, r, s, t } is a lattice represented by the following Hasse diagram: For any $$x, y ∈ L$$, not necessarily distinct, $$x ∨ y$$ and x ∧ y are join and meet of x, y, respectively. Let $$L^3 = \left\{\left(x, y, z\right): x, y, z ∈ L\right\}$$ be the set of all ordered triplets of the elements of L. Let p_r be the probability that an element $$\left(x, y,z\right) ∈ L^3$$ chosen equiprobably satisfies $$x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)$$. Then

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 ________.