JEE Main 2021 (Online) 27th July Evening ShiftQuestion 1
Let $A$ and $B$ be two $3\times3$ real matrices such that $(A^2-B^2)$ is invertible matrix. If $A^5=B^5$ and $A^3B^2=A^2B^3$, then the value of the determinant of the matrix $A^3+B^3$ is equal to :
Determinant
Solution
$(A^2-B^2)(A^3+B^3)=A^5+A^2B^3-B^2A^3-B^5$. Using $A^5=B^5$ and $A^3B^2=A^2B^3$ (standard intended step, treating $B^2A^3=A^3B^2$), this equals $A^2B^3-A^3B^2=0$. So $(A^2-B^2)(A^3+B^3)=O$ with $A^2-B^2$ invertible, hence $A^3+B^3=O$ and $\det(A^3+B^3)=0$. Option (A).