Question

Suppose that A and B are arbitrary 2×22 \times 2 matrices. Is the quantity given always equal to (A+B)2?(A+B)^{2} ? (B+A)2(B+A)^{2}

Solution

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

(A+B)2=A(A+B)+B(A+B)=A2+AB+BA+B2        (1)(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)=B2+BA+AB+A2        (2)(B+A)^2 = B(B+A)+B(B+A)=B^2+BA+AB+A^2\;\;\;\;(2)

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

A2+AB+BA+B2=B2+BA+AB+A2A^2+AB+BA+B^2 = B^2+BA+AB+A^2

from the commutativity of addition. Hence, (A+B)2=(B+A)2=A2+AB+BA+B2(A+B)^2=(B+A)^2=A^2+AB+BA+B^2. They are equal.

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