Step 1: Find the eigenvalues and eigenvectors of A = [[4, 1], [2, 3]]. Step 2: Solve the characteristic equation det(A − λI) = 0: |4−λ, 1; 2, 3−λ| = (4−λ)(3−λ) − 2 = 0. Step 3: Expand: λ² − 7λ + 12 − 2 = λ² − 7λ + 10 = 0. Factor: (λ−5)(λ−2) = 0. Step 4: Eigenvalues: λ_1 = 5, λ_2 = 2. Check: trace = 7 = 5+2 ✓; determinant = 12−2 = 10 = 5×2 ✓. Step 5: For λ_1 = 5: (A − 5I)v = 0 gives −x + y = 0, so v_1 = (1, 1). Step 6: For λ_2 = 2: (A − 2I)v = 0 gives 2x + y = 0, so v_2 = (1, −2). Result: A stretches vectors along (1,1) by factor 5 and along (1,−2) by factor 2. Then A = PDP⁻¹ with P = [[1,1],[1,−2]], D = diag(5,2).