Exercise 1: Finding Eigenvalues and Eigenvectors of C — Possible Solution
====================================================================

GIVEN
------------------------------
C = [[4, 2], [1, 3]]

STEP 1: THE CHARACTERISTIC EQUATION
------------------------------
trace(C) = 4 + 3 = 7
det(C)   = (4)(3) - (2)(1) = 12 - 2 = 10

Per this chapter's own formula: L^2 - trace(C)L + det(C) = 0

  L^2 - 7L + 10 = 0
  (L - 2)(L - 5) = 0
  L = 2  or  L = 5

Check: sum = 2 + 5 = 7 = trace ✓. Product = 2 x 5 = 10 = det ✓.

STEP 2: EIGENVECTOR FOR L = 2
------------------------------
C - 2I = [[4-2, 2], [1, 3-2]] = [[2, 2], [1, 1]]

Row 1 gives: 2v1 + 2v2 = 0  ->  v2 = -v1

Eigenvector direction: [1, -1]

STEP 3: EIGENVECTOR FOR L = 5
------------------------------
C - 5I = [[4-5, 2], [1, 3-5]] = [[-1, 2], [1, -2]]

Row 1 gives: -v1 + 2v2 = 0  ->  v1 = 2v2

Eigenvector direction: [2, 1]

STEP 4: VERIFYING BOTH EIGENPAIRS
------------------------------
C [1, -1] = [(4)(1) + (2)(-1), (1)(1) + (3)(-1)] = [4-2, 1-3] = [2, -2]
          = 2 x [1, -1]  ✓  matches L = 2

C [2, 1]  = [(4)(2) + (2)(1), (1)(2) + (3)(1)]   = [8+2, 2+3]   = [10, 5]
          = 5 x [2, 1]   ✓  matches L = 5

WHY THIS WORKS AS AN ANSWER
------------------------------
Both eigenvalues come from this chapter's own characteristic-equation
formula built from the trace and determinant, each eigenvector is
found by solving (C - LI)v = 0 using this chapter's own Chapter-6-
style row equation, and both results are independently confirmed by
directly computing C v and comparing it to L v, rather than trusting
the algebra alone.