We're going to consider the matrix
\(\begin{pmatrix} -4 & -15 \\ 2 & 7 \end{pmatrix}\)
a)
Let \({P} = \begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix}\).
Find the 2x2 matrix D such that
\({P} {D} {P}^{-1} = \begin{pmatrix} -4 & -15 \\ 2 & 7 \end{pmatrix}\)
\(\text{Let $M = \begin{pmatrix} -4 & -15 \\ 2 & 7 \end{pmatrix} $ }\)
Formula:
\(\begin{array}{|rcll|} \hline P^{-1} &=& \dfrac{1}{\begin{vmatrix} 3 & -5 \\ -1 & 2 \end{vmatrix}}\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} \\\\ &=& \dfrac{1}{6-5}\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} \\\\ &=& \dfrac{1}{1}\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} \\\\ \mathbf{P^{-1}} & \mathbf{=} & \mathbf{\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline PDP^{-1} &=& M \quad | \quad \cdot P \\ PDP^{-1}P &=& MP \quad | \quad P^{-1}P = I(\text{Identity matrix}) \\ PDI &=& MP \quad | \quad \cdot P^{-1} \\ P^{-1}PDI &=& P^{-1}MP \quad | \quad P^{-1}P = I(\text{Identity matrix}) \\ IDI &=& P^{-1}MP \\ \mathbf{D} & \mathbf{=} & \mathbf{P^{-1}MP} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{D} & \mathbf{=} & \mathbf{P^{-1}MP} \\ &=& \begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}\begin{pmatrix} -4 & -15 \\ 2 & 7 \end{pmatrix}\begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix} \\ &=& \begin{pmatrix} 2 & 5 \\ 2 & 6 \end{pmatrix}\begin{pmatrix} 3 & -5 \\ -1 & 2 \end{pmatrix} \\ \mathbf{D} & \mathbf{=} & \mathbf{\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}} \quad | \quad (\text{Diagonal matrix}) \\ \hline \end{array}\)