1.4.17 Matrix transformations

condition: Find the eigenvalues ​​and eigenvectors of the linear operator given by the matrix \(A\). Prove that this is an operator of a simple type, reduce its matrix to diagonal form (find the transition matrix to its own basis and check). Compute \( A^{n} \) for any \( n \in \mathbb{N} \). \[ A=\left(\begin{array}{ll} -4 & 5 \\ -1 & 2 \end{array}\right) \]