First, let's find the eigenvalues of matrix A.

Eigenvalues are found by solving the characteristic equation, which is given by:

det(A - λI) = 0

where A is the matrix, λ is the eigenvalue, and I is the identity matrix.

For matrix A:

2 2 6
2 10 2
6 2 2

The characteristic equation becomes:

det(A - λI) = (2 - λ)(10 - λ)(2 - λ) + 2(2 - λ)(6 - 2λ) + 6(2λ - 6) = 0

Expanding and simplifying the equation gives:

(λ - 2)(λ - 10)(λ - 2) + 2(λ - 2)(-λ + 6) + 6(2λ - 6) = 0

(λ - 2)^2(λ - 10) - 2(λ - 2)(λ - 6) + 12(λ - 2) = 0

Simplifying further:

(λ - 2)[(λ - 2)(λ - 10) - 2(λ - 6) + 12] = 0

Now, we have two factors: (λ - 2) = 0 and [(λ - 2)(λ - 10) - 2(λ - 6) + 12] = 0

Solving the first factor, we get:

λ - 2 = 0
λ = 2

Now, we solve the second factor:

(λ - 2)(λ - 10) - 2(λ - 6) + 12 = 0
(λ^2 - 10λ - 2λ + 20) - (2λ - 12) + 12 = 0
λ^2 - 12λ + 20 - 2λ + 12 + 12 = 0
λ^2 - 14λ + 44 = 0

Using the quadratic formula, we can solve for λ:

λ = (-(-14) ± √((-14)^2 - 4(1)(44))) / (2(1))
λ = (14 ± √(196 - 176)) / 2
λ = (14 ± √20) / 2
λ = (14 ± 2√5) / 2
λ = 7 ± √5

Therefore, the eigenvalues of matrix A are 2, 7 + √5, and 7 - √5.

Now, let's calculate the values of pq, pr, and rq - pqr.

- pq = (2)(7 + √5) = 14 + 2√5
- pr = (2)(7 - √5) = 14 - 2√5
- rq - pqr = (7 + √5)(7 - √5) - (2)(7 + √5)(7 - √5) = 49 - 5 - 14√5 + 14√5 = 44

Therefore, pq = 14 + 2√5, pr = 14 - 2√5, and rq -