Question

Let $A$ be a square matrix of order $n\times n$ and let $P$ be a non-singular matrix, then which of the following matrices have the same characteristic roots.

• A. $A$ and $PA$
• B. $A$ and $AP$
• C. $A$ and $P^{-1}AP$
• D. none of these

Answer 1

The correct option is C $A$ and $P^{-1}AP$

Let the characteristic root of $A$ be $\lambda$.

$\Rightarrow |A-\lambda I|=0$

For $P^{-1}AP;|P^{-1}AP-\lambda I|=|P^{-1}AP-\lambda P^{-1}PI|=\left|P^{-1}\right||A-\lambda I|\left|P\right|=|A-\lambda 1|.$

$\Rightarrow \lambda$ is also characteristic roots of matrix $P^{-1}AP$