Question

# Suppose that A and B are arbitrary $2 \times 2$ matrices. Is the quantity given always equal to $(A+B)^{2} ?$ $(B+A)^{2}$

Solution

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Step 1
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Recall that while multiplication is not commutative, addition is commutative. Hence $A+B=B+A$. By applying the distributive properties, we can see that:

$(A+B)^2 =A(A+B)+B(A+B)=A^2+AB+BA+B^2 \;\;\;\;(1)$

$(B+A)^2 = B(B+A)+B(B+A)=B^2+BA+AB+A^2\;\;\;\;(2)$

Observe equations $(1)$ and $(2)$. We see that:

$A^2+AB+BA+B^2 = B^2+BA+AB+A^2$

from the commutativity of addition. Hence, $(A+B)^2=(B+A)^2=A^2+AB+BA+B^2$. They are equal.

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