###### Exercise 3

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

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

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

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

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

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