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Algebra

Matrices, Cayley-Hamilton Theorem

jee_main_2026_april_2_shift_1

Grade 12

Question:

If A = [[1, 2], [3, 4]], then the value of A^2 - 5A - 2I is:

A. Zero matrix

B. Identity matrix

C. [[2, 0], [0, 2]]

D. [[0, 2], [2, 0]]

Step-by-Step Solution

Key Concept: Use the Cayley-Hamilton theorem or compute directly.

Step 1: A^2 = [[1,2],[3,4]] × [[1,2],[3,4]] = [[7,10],[15,22]]. Step 2: 5A = [[5,10],[15,20]], 2I = [[2,0],[0,2]]. Step 3: A^2 - 5A - 2I = [[7-5-2, 10-10-0], [15-15-0, 22-20-2]] = [[0,0],[0,0]].

Correct Answer: A

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