Exercise 3

Determine whether the following statements are true or false and provide a justification for your response.

  1. If \(A\) is invertible, then \(A\) is diagonalizable.

  2. If \(A\) and \(B\) are similar and \(A\) is invertible, then \(B\) is also invertible.

  3. If \(A\) is a diagonalizable \(n\times n\) matrix, then there is a basis of \(\real^n\) consisting of eigenvectors of \(A\text{.}\)

  4. If \(A\) is diagonalizable, then \(A^{10}\) is also diagonalizable.

  5. If \(A\) is diagonalizable, then \(A\) is invertible.