Linear Algebra with Applications, 10th Edition

Section 1.6 Exercises

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

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

   2. $\begin{array}{cc}\left[\begin{array}{c}A\\ I\end{array}\right]& A^{-1}\end{array}$

   3. $\begin{array}{cc}[\begin{array}{cc}A& I\end{array}{]}^T& \left[\begin{array}{cc}A& I\end{array}\right]\end{array}$

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

   5. $\begin{array}{cc}\left[\begin{array}{c}{A}^{-1}\\ I\end{array}\right]& \left[\begin{array}{cc}A& I\end{array}\right]\end{array}$

2. Let $B=A^{T}A$. Show that $b_{ij}={\mathbf{a}}_i^T{\mathbf{a}}_j$.

3. Let

   $$\begin{array}{cc}\begin{array}{cc}A=\left[\begin{array}{cc}1& \hfill 1\\ 2& \hfill -1\end{array}\right]& \text{ and }\end{array}& B=\left[\begin{array}{cc}2& 1\\ 1& 3\end{array}\right]\end{array}$$

   1. Calculate $A{\mathbf{b}}_1$ and $A{\mathbf{b}}_2$.

   2. Calculate ${\overrightarrow{\mathbf{a}}}_{1}B$ and ${\overrightarrow{\mathbf{a}}}_{2}B$.

   3. Multiply AB and verify that its column vectors are the vectors in part (a) and its row vectors are the vectors in part (b).

4. Let

   $$\begin{array}{l}I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],E=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],O=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\\ C=\left[\begin{array}{cc}\hfill 1& 0\\ \hfill -1& 1\end{array}\right],D=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]\end{array}$$

   and

   

   Perform each of the following block multiplications:

   1. $\begin{array}{cc}\left[\begin{array}{cc}O& I\\ I& O\end{array}\right]& \left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right]\end{array}$

   2. $\begin{array}{cc}\left[\begin{array}{cc}C& O\\ O& C\end{array}\right]& \left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right]\end{array}$

   3. $\begin{array}{cc}\left[\begin{array}{cc}D& O\\ O& I\end{array}\right]& \left[\begin{array}{c}\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\end{array}\right]\end{array}$

   4. $\begin{array}{cc}\left[\begin{array}{cc}E& O\\ O& E\end{array}\right]& \left[\begin{array}{c}\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\end{array}\right]\end{array}$

5. Perform each of the following block multiplications:

   1.