Q. 31

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

Answer :

By matrix multiplication we can write:

(A + B)^{2} = (A+B)(A+B) = A^{2} + AB + BA + B^{2}

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

∴ (A + B)^{2} = A^{2} + AB + AB + B^{2}

⇒ (A + B)^{2} = A^{2} + 2AB + B^{2} …proved

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