Given the matrix \(P = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix},\) we want to find \( P^2 - 4P - I \), where \( I \) is the identity matrix: \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.\) Calculate {\( P^2 \)} \(P^2 = P \times P = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix}.\) Calculate {\( 4P \)} \(4P = 4 \times \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 4 & 8 \\ 8 & 12 \end{bmatrix}\) Calculate {\( P^2 - 4P \)} \(P^2 - 4P = \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix} - \begin{bmatrix} 4 & 8 \\ 8 & 12 \end{bmatrix} = \begin{bmatrix} 5 - 4 & 8 - 8 \\ 8 - 8 & 13 - 12 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) Calculate {\( P^2 - 4P - I \)} \(P^2 - 4P - I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\) Thus, the result of \( P^2 - 4P - I \) is \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\)