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Let A and B be square matrices of the order 3 × 3. Is (AB)2 = A2B2 ? Give reasons.

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We are given that,

A and B are square matrices of the order 3 × 3.

We need to check whether (AB)2 = A2B2 is true or not.

Take (AB)2.

(AB)2 = (AB)(AB)

[∵ A and B are of order (3 × 3) each, A and B can be multiplied; A and B be any matrices of order (3 × 3)]

⇒ (AB)2 = ABAB

[∵ (AB)(AB) = ABAB]

⇒ (AB)2 = AABB

[∵ ABAB = AABB; as A can be multiplied with itself and B can be multiplied by itself]

⇒ (AB)2 = A2B2

So, note that, (AB)2 = A2B2 is possible.

But this is possible if and only if BA = AB.

And BA = AB is always true whenever A and B are square matrices of any order. And for BA = AB,

(AB)2 = A2B2

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