Exercise 14

We can ask how the matrix product operation interacts with the transpose operation.

• Show that ${{(GH)}^{\mathsf{T}}}={{H}^{\mathsf{T}}}{{G}^{\mathsf{T}}}$.

• A square matrix is symmetric if each $i,j$ entry equals the $j,i$ entry, that is, if the matrix equals its own transpose. Show that the matrices $H{{H}^{\mathsf{T}}}$ and ${{H}^{\mathsf{T}}}H$ are symmetric.