Monday, November 14th

Problem
Let A and B be different nxn matrices with real entries. If $A^3=B^3$ and $A^2B=AB^2$, can $A^2+B^2$ be invertible? 


Solution
Since $(A-B)(A^2+B^2)=A^3+AB^2-BA^2-B^3=A^3-B^3=0$ and $A-B\neq 0$, we have that $A^2+B^2=0$ and hence, it is not invertible.