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Question
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.
Options
Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
Both Assertion and Reason are true but Reason is not the correct explanation for Assertion.
Assertion is true and Reason is false.
Assertion is false and Reason is true.
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Solution
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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