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.

13.

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\mathbf{}\left[\begin{array}{c}7\\ 11\end{array}\right]$

Short answer

The dot product$\mathrm{A}\overrightarrow{\mathrm{x}}\mathrm{of}\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{c}7\\ 11\end{array}\right]\mathrm{is}\left[\begin{array}{c}29\\ 65\end{array}\right]$

Step-by-step solution

Step 1:

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

$$\begin{gathered} \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{c}7\\ 11\end{array}\right] \\ \Rightarrow 7\left[\begin{array}{c}1\\ 3\end{array}\right]+11\left[\begin{array}{c}2\\ 4\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}7\\ 21\end{array}\right]+\left[\begin{array}{c}22\\ 44\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}7+22\\ 21+44\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}29\\ 65\end{array}\right] \end{gathered}$$

:

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

$$\begin{gathered} \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{c}7\\ 11\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}\left(1\right)\left(7\right)+\left(2\right)\left(11\right)\\ \left(3\right)\left(7\right)+\left(4\right)\left(11\right)\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}7+22\\ 21+44\end{array}\right] \\ \Rightarrow \left[\begin{array}{c}29\\ 65\end{array}\right] \end{gathered}$$