Question

If $A$ and $B$ are different matrices satisfying $A^3=B^3\text{and}{A}^{2}B=B^{2}A,$ then

• A. ${\text{det(A}}^2+{\text{B}}^2)\text{must be zero}$
• B. $\text{det (A-B) must be zero}$
• C. Both ${\text{det(A}}^2+{\text{B}}^2)$ &$\text{det(A-B) must be zero}$
• D. ${\text{At least one of det (A}}^2+{\text{B}}^2)\text{or det (A-B) must be zero}$

Answer 1

The correct option is D ${\text{At least one of det (A}}^2+{\text{B}}^2)\text{or det (A-B) must be zero}$

$A^3=B^3\cdots \left(i\right)$ and
$A^{2}B=B^{2}A\cdots \left(ii\right)$on subtracting
equation $\left(i\right)$ and $\left(ii\right)$ we get
$A^3-A^{2}B=B^3-B^{2}A\Rightarrow A^2(A-B)=-B^2(A-B)$
$\Rightarrow (A^2+B^2)(A-B)=O$
$\Rightarrow |(A^2+B^2).(A-B)|=0$
$\Rightarrow \text{det}(A^2+B^2)=0$ or $\text{det}(A-B)=0$