Chapter 1.6, Problem 1E

Let A be a nonsingular $n\times n$ matrix. Perform the following multiplications:

(a) $A^{-1}\left[\begin{array}{cc}A& I\end{array}\right]$

(b) $\left[\begin{array}{l}A\\ I\end{array}\right]A^{-1}$

(c) ${\left[\begin{array}{cc}A& I\end{array}\right]}^T\left[\begin{array}{cc}A& I\end{array}\right]$

(d) $\left[\begin{array}{cc}A& I\end{array}\right]{\left[\begin{array}{cc}A& I\end{array}\right]}^T$

(e) $\left[\begin{array}{l}{A}^{-1}\\ I\end{array}\right]\left[\begin{array}{cc}A& I\end{array}\right]$