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.

Answer 1

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.