Matrices A and B are invertible and have the same eigenvector v for different corresponding eigenvalues. Show that the inverse of (AB) also has eigenvector v and find the corresponding eigenvalue.
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Best Answer
Hint. So, let's say $Av = \lambda v$, $Bv = \mu v$. Now compute $$ (AB)v = A(Bv) = A(\mu v) = \mu Av = \ldots $$ and multiply both sides by $(AB)^{-1}$.