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 $A$.

4. If $A$ is diagonalizable, then $A^{10}$ is also diagonalizable.

5. If $A$ is diagonalizable, then $A$ is invertible.