(a) Suppose that A is a 3 x 3 diagonalizable matrix with a

Last updated: 8/11/2022

(a) Suppose that A is a 3 x 3 diagonalizable matrix with a basis of eigenvectors of R³ given by {u₁, U₂, U3} with corresponding eigenvalues A1, A2, A3. Suppose that the matrix B is also a 3 x 3 diagonalizable matrix with the same eigenvectors although with possibly different eigenvalues 11, 12, 13. Prove that A + B is also a diagonalizable matrix. (b) Suppose that the square matrices A and B are similar. Prove that the matrices A2+ A+ I and B2 +B+ I are also similar. (c) Suppose that the square matrices A and B both have their characteristic polynomial equal to -³ + X. Prove that A is similar to B.