Let A and B be two 33 real matrices such that (A^2-B^2) is an invertible matrix.

Question

Let $A$ and $B$ be two $3	imes3$ real matrices such that $(A^2-B^2)$ is an 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:

Accepted Answer

From $A^5=B^5$ and $A^3B^2=A^2B^3$: $A^5-A^3B^2=B^5-A^2B^3 \Rightarrow A^3(A^2-B^2)+B^3(A^2-B^2)=0 \Rightarrow (A^3+B^3)(A^2-B^2)=O$.

Since $A^2-B^2$ is invertible, post-multiplying by its inverse gives $A^3+B^3=O$, so $\det(A^3+B^3)=0$.

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