Show that if A and B are square matrices

Question:

Show that if A and B are square matrices such that AB = BA, then

(A + B)2 = A2 + 2AB + B2.

Solution:

Given, A and B are square matrices such that AB = BA.

So, (A + B)2 = (A + B) . (A + B)

= A2 + AB + BA + B2

= A2 + AB + AB + B2 [Since, AB = BA]

= A+ 2AB + B2

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