Show that if A and B are square matrices such that AB = BA, then (A + B)² = A² + 2AB + B².

Question

Show that if A and B are square matrices such that AB = BA, then (A + B) 2 = A 2 + 2AB + B 2 .

Philoid · Accepted Answer

By matrix multiplication we can write:

(A + B)² = (A+B)(A+B) = A² + AB + BA + B²

We know that matrix multiplication is not commutative but it is given that : AB = BA

∴ (A + B)² = A² + AB + AB + B²

⇒ (A + B)² = A² + 2AB + B² …proved

Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2.