Exercise 10

The equation ending Example 2.5

is a bit jarring because for $P$ we must take the first matrix, which is shown as an inverse, and for $P^{-1}$ we take the inverse of the first matrix, so that the two $-1$ powers cancel and this matrix is shown without a superscript $-1$.

• Check that this nicer-appearing equation holds.

  $$\begin{pmatrix}3&0\\ 0&1\end{pmatrix}=\begin{pmatrix}1&1\\ 0&-1\end{pmatrix}\begin{pmatrix}3&2\\ 0&1\end{pmatrix}\begin{pmatrix}1&1\\ 0&-1\end{pmatrix}^{-1}$$

• Is the previous item a coincidence? Or can we always switch the $P$ and the $P^{-1}$?