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