## Linear Algebra and Its Applications, exercise 1.4.19

Exercise 1.4.19. Given matrices A and B, which of the following matrices are equal to $(A + B)^2$?

1. $(B + A)^2$
2. $A^2 + 2AB + B^2$
3. $A(A + B) + B(A + B)$
4. $(A + B)(B + A)$
5. $A^2 + AB + BA + B^2$ $A + B = B + A \Rightarrow (A + B)^2 = (B + A)^2$

as well as $A + B = B + A \Rightarrow (A + B)^2 = (A + B)(A + B) = (A + B)(B + A)$

So the matrices referenced in (a) and (d) above are equal to the original matrix $(A + B)^2$.

Second, since matrix multiplication is distributive we have $(A + B)^2 = (A + B)(A + B) = A(A + B) + B(A + B)$

and $A(A + B) + B(A + B) = A^2 + AB + BA + B^2$

So the matrices referenced in (c) and (e) above are equal to the original matrix $(A + B)^2$.

However in general matrix multiplication is not commutative, so $AB \ne BA \Rightarrow A^2 + AB + BA + B^2 \ne A^2 + 2AB + B^2$

and the matrix referenced in item (b) above is not in general equal to $(A + B)^2$.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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