Exercise 2: Checking a Claimed Eigenpair — Possible Solution
====================================================================

GIVEN
------------------------------
D = [[5, -2], [1, 2]]
Claimed eigenpair: L = 4, v = [2, 1]

STEP 1: DIRECT CHECK — COMPUTING D v
------------------------------
D v = [(5)(2) + (-2)(1), (1)(2) + (2)(1)] = [10 - 2, 2 + 2] = [8, 4]

STEP 2: COMPARING TO 4v
------------------------------
4v = 4 x [2, 1] = [8, 4]

D v = [8, 4] and 4v = [8, 4] - they match exactly, so the claimed
eigenpair does satisfy this chapter's own defining equation, A v = L v.

STEP 3: INDEPENDENT VERIFICATION VIA THE CHARACTERISTIC EQUATION
------------------------------
trace(D) = 5 + 2 = 7
det(D)   = (5)(2) - (-2)(1) = 10 + 2 = 12

Characteristic equation: L^2 - 7L + 12 = 0
  (L - 3)(L - 4) = 0
  L = 3  or  L = 4

4 does appear as one of the two eigenvalues produced by this chapter's
own characteristic-equation formula, confirming the colleague's claim
a second, independent way.

WHY THIS WORKS AS AN ANSWER
------------------------------
The claim is checked twice by two genuinely different methods - a
direct computation of D v compared against 4v, and an independent
derivation of D's actual eigenvalues from its trace and determinant -
rather than accepting the claim on the strength of only one check, or
assuming it's correct because the numbers "look plausible."