Question

If $A$ and $B$ are two distinct matrices such that $A^3=B^3$ and $A^{2}B=B^{2}A$, find $det(A^2+B^2)$.

Answer 1

$A^3=B^3,A^{2}B=B^{2}A$ -----Given

$\Rightarrow A^{2}B-B^{2}A=0$

$A^3=B^3$ ----Given

$\Rightarrow (A^2+B^2)(A-B)=A^3-A^{2}B+B^{2}A-B^3$

$=(A^3-B^3)-A^{2}B-B^2$

$=0$

$\because A\ne 3$

$\Rightarrow A^2+B^2=0$

$det=0$