Exercise 3

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

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

2. Suppose that $A$ is a $3\times3$ matrix and that there is a matrix $P$ such that $$A = P \left[\begin{array}{rrr} 2 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & -3 \\ \end{array}\right] P^{-1}$$. Find $\det(A)$.