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      Question

      Given matrix A = [[3, 1], [2, 4]], evaluate A² − 5A +

      5I (where I is the 2×2 identity matrix):
      A [[1, 2], [4, 3]] Correct Answer Incorrect Answer
      B [[0, 0], [0, 0]] — the zero matrix Correct Answer Incorrect Answer
      C [[2, 0], [0, 2]] Correct Answer Incorrect Answer
      D [[−2, 0], [0, −2]] Correct Answer Incorrect Answer

      Solution

      Step 1 — A² = A × A: Row 1: [(3×3)+(1×2), (3×1)+(1×4)] = [11, 7] Row 2: [(2×3)+(4×2), (2×1)+(4×4)] = [14, 18] So A² = [[11, 7], [14, 18]] Step 2 — 5A = [[15, 5], [10, 20]] Step 3 — 5I = [[5, 0], [0, 5]] Step 4 — A² − 5A + 5I = [[11−15+5, 7−5+0], [14−10+0, 18−20+5]] = [[1, 2], [4, 3]] Note: Cayley-Hamilton Theorem: Every matrix satisfies its own characteristic equation. For this A, the characteristic equation is λ² − 7λ + 10 = 0, meaning A² − 7A + 10I = 0 (the zero matrix). If the exam expression were A² − 7A + 10I, the answer would be the zero matrix. Always check which expression is given.

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