Is binomial expansion true for all kinds of matrices

Question

My teacher told me that, in case of square matrices, binomial expansion holds true if and only if they commute. I am not able to figure out why they fail for other cases; any proof will be helpful.

Answer 1

Depends what you mean. $$(A+B)^3 = A^3 + 3 A^2 B + 3 A B^2 + B^3$$ also works for $$ A = \pmatrix{a_{11} & 0\cr a_{21} & a_{22}\cr},\ B = \pmatrix{b_{11} & 0\cr b_{21} & -a_{11}-2 a_{22}-2b_{11}\cr} $$ which in general do not commute.

Answer 2

Hint:

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