Chapter 1.6, Problem 40E

The matrices and vectors listed in Eq. (3) are used in several of the exercises that follow.

$A=\left[\begin{array}{cc}3& 1\\ 4& 7\\ 2& 6\end{array}\right]$, $B=\left[\begin{array}{ccc}1& 2& 1\\ 7& 4& 3\\ 6& 0& 1\end{array}\right]$

$C=\left[\begin{array}{cccc}2& 1& 4& 0\\ 6& 1& 3& 5\\ 2& 4& 2& 0\end{array}\right]$, $D=\left[\begin{array}{cc}2& 1\\ 1& 4\end{array}\right]$

$E=\left[\begin{array}{cc}3& 6\\ 2& 3\end{array}\right]$, $F=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

$\mathbf{u}=\left[\begin{array}{c}1\\ -1\end{array}\right]$, $\mathbf{v}=\left[\begin{array}{c}-3\\ 3\end{array}\right]$ (3)

If $\mathbf{x}$ and y are vectors in $R^n$, then the product ${\mathbf{x}}^T\mathbf{y}$ is often called an inner product. Similarly, the product $\mathbf{x}{\mathbf{y}}^{\mathrm{T}}$ is often called an outer product. Exercises 35-40 concern outer products; the matrices and vectors are given in Eq. (3). In Exercises 35-40, form the outer products.

${A\mathbf{v}\left(\mathrm{A}\mathbf{u}\right)}^T$