Question

Compute the products Axin Exercises 13 through 15 using

paper and pencil. In each case, compute the product

two ways: in terms of the columns of A and in terms of the rows of A.

14.

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\end{array}\right]\left[\begin{array}{c}-1\\ 2\\ 1\end{array}\right]$

Short answer

The dot product$A{x}^{\to }\mathrm{of}\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\end{array}\right]\left[\begin{array}{c}-1\\ 2\\ 1\end{array}\right]\mathrm{is}\left[\begin{array}{c}6\\ 8\end{array}\right]$

Step-by-step solution

Step 1:

Find$A\overline{x}.$in terms of rows of A.

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\end{array}\right]\left[\begin{array}{c}-1\\ 2\\ 1\end{array}\right]$

$$\begin{gathered} \Rightarrow \left[\begin{array}{c}\left(1\right)\left(-1\right)+\left(2\right)\left(2\right)+\left(3\right)\left(1\right)\\ \left(2\right)\left(-1\right)+\left(3\right)\left(2\right)+\left(4\right)\left(1\right)\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}-1+4+3\\ -2+6+4\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}6\\ 8\end{array}\right] \end{gathered}$$

Step 2:

Find $A\overline{x}.$in terms of columns of A.

$\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\end{array}\right]\left[\begin{array}{c}-1\\ 2\\ 1\end{array}\right]$

$$\begin{gathered} \Rightarrow -1\left[\begin{array}{c}1\\ 2\end{array}\right]+2\left[\begin{array}{c}2\\ 3\end{array}\right]+1\left[\begin{array}{c}3\\ 4\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}-1+4+3\\ -2+6+4\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}6\\ 8\end{array}\right] \end{gathered}$$