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

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

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

Answer»

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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