Exercise 7

Explain why the following statements are true.

1. Given any matrix $B$, the matrix $B^TB$ is a symmetric, positive semidefinite matrix.

2. If both $A$ and $B$ are symmetric, positive definite matrices, then $A+B$ is a symmetric, positive definite matrix.

3. If $A$ is a symmetric, invertible, positive definite matrix, then $A^{-1}$ is also.