##### Exercise3

In the next chapter, we will say that matrices $$A$$ and $$B$$ are similar if there is a matrix $$P$$ such that $$A= PBP^{-1}\text{.}$$

1. Suppose that $$A$$ is a $$3\times3$$ matrix and that there is a matrix $$P$$ such that

\begin{equation*} A = P \left[\begin{array}{rrr} 2 \amp 0 \amp 0 \\ 0 \amp -5 \amp 0 \\ 0 \amp 0 \amp -3 \\ \end{array}\right] P^{-1} \text{.} \end{equation*}

Find $$\det A\text{.}$$

2. Suppose that $$A$$ and $$B$$ are matrices and that there is a matrix $$P$$ such that $$A=PBP^{-1}\text{.}$$ Explain why $$\det A = \det B\text{.}$$

in-context