Exercise 4.4.18

Avg difficulty 3.0

Suppose $A$ and $B$ are square matrices of the same size. Which of the following are necessarily true?

$(A+B)^{2}=A^{2}+2AB+B^{2}$ .
$(A-B)^{2}=A^{2}-2AB+B^{2}$ .
$(AB)^{2}=A^{2}B^{2}$ .
$(A+B)^{2}=A^{2}+AB+BA+B^{2}$ .
$A^{2}B^{2}=A(AB)B$ .
$(A+B)^{3}=A^{3}+3A^{2}B+3AB^{2}+B^{3}$ .
$(A+B)(A-B)=A^{2}-B^{2}$ .

Selinger, Matrix Theory and Linear Algebra, Matrices, Matrix multiplication, Exercise 4.4.18

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.