For the matrix A , there is an invertible matrix P such that P^(−1)*AP = D, where D is a diagonal matrix with the eigenvalues of A on the diagonal. Calculate the matrix product P^(-1)*AP

If A is a 3x3 matrix with eigenvalues 1 and​ 2, which option is impossible for diagonal matrix​ D=P^{-1}AP? a. D=diag(1,2,2) b. ​D=diag(1,1,1) c. D=diag(1,1,2) d. D does not exist

Find e^(tA). Use e^(tA) = P e^(tD) P^(-1). The matrix is \begin{pmatrix}1&2&0\\ 0&1&-2\\ 2&2&-1\end{pmatrix}. Use det(A-λI) for eigen values and the find eigen vector using Av = λv. For the eigen vector : \begin{pmatrix}1&2&0\\ 0&1&-2\\ 2&2&-1\end{pmatrix}\cdot \begin{pmatrix}x\\ y\\ z\end{pmatrix}=λ\cdot \begin{pmatrix}x\\ y\\ z\end{pmatrix}. After finding the eigen values, use this to find eigen vectors. Then find e^(tA). use e^(tA) = P e^(tD) P^(-1).

(9) If 2 0 0 1 3 A-'B-1 and A = 2 -1 evaluate the matrix B.