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

Answer: First, since matrix addition is commutative we have

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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