Assertion: Let the matrices A = ((-3, 2),(-5, 4)) and B = ((4, -2),(5, -3)) be such that A100B = BA100 Reason: AB = BA implies AB = BA for all positive integers n.

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Both Assertion and Reason are true and Reason is the correct explanation for Assertion. Explanation: We have, A = [(-3, 2),(-5, 4)], B = [(4, -2),(5, -3)] Now, AB = `[(-3, 2),(-5, 4)][(4, -2),(5, -3)] = [(-2, 0),(0, -2)] And BA = [(4, -2),(5, -3)][(-3, 2),(-5, 4)] = [(-2, 0),(0, -2)] Hence, AB = BA Now, A2 = [(-3, 2),(-5, 4)][(-3, 2),(-5, 4)] = [(-1, 2),(-5, 6)] So, A2B = [(-1, 2),(-5, 6)][(4, -2),(5, -3)] = [(6, -4),(10, -8)] And BA2 = [(4, -2),(5, -3)][(-1, 2),(-5, 6)] = [(6, -4),(10, -8)]` Hence, A2B = BA2 If, AB = BA and A2B = BA2............... Therefore, AnB = BAn Also, A100B = BA100 Hence, Assertion and Reason both are true.

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