Is there any relation between index of matrix $AB$ and matrices $ A$, and $B$? The index of a matrix $A$ is the smallest nonnegative integer $k$ such that $\operatorname{rank}(A^{k+1})=\operatorname{rank}(A^k)$.

Now I want to know: what relation is there between $\operatorname{Index}(AB)$ and $\operatorname{Index}(A)$, $\operatorname{Index}(B)$?
No it is not true. Consider matrices $A=\begin{pmatrix}1&amp; 0\\  0 &amp; 0\end{pmatrix}$ and $A'=\begin{pmatrix}0&amp; 1\\  0 &amp; 0\end{pmatrix}$, the index of $A$ is one and the index of $A'$ is 2. You are right if the matrices are nonsingular.
No it is not true. Consider matrices $A=\begin{pmatrix}1&amp; 0\\  0 &amp; 0\end{pmatrix}$ and $A'=\begin{pmatrix}0&amp; 1\\  0 &amp; 0\end{pmatrix}$, the index of $A$ is one and the index of $A'$ is 2. You are right if the matrices are nonsingular.