Exercise 8

Determine whether the following statements are true or false and explain your thinking.

1. If $A$ is an invertible, orthogonally diagonalizable matrix, then so is $A^{-1}$.

2. If $\lambda=2+i$ is an eigenvalue of $A$, then $A$ cannot be orthogonally diagonalizable.

3. If there is a basis for $\real^m$ consisting of eigenvectors of $A$, then $A$ is orthogonally diagonalizable.

4. If $\uvec$ and $\vvec$ are eigenvectors of a symmetric matrix associated to eigenvalues -2 and 3, then $\uvec\cdot\vvec=0$.

5. If $A$ is a square matrix, then $\uvec\cdot(A\vvec) = (A\uvec)\cdot\vvec$.