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Q13E

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Found in: Page 35

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Compute the products Ax in Exercises 13 through 15 usingpaper 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.$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\mathbf{}\left[\begin{array}{c}7\\ 11\end{array}\right]$

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

See the step by step solution

## Step 1:

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

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{c}7\\ 11\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒7\left[\begin{array}{c}1\\ 3\end{array}\right]+11\left[\begin{array}{c}2\\ 4\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{c}7\\ 21\end{array}\right]+\left[\begin{array}{c}22\\ 44\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{c}7+22\\ 21+44\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{c}29\\ 65\end{array}\right]$

## Step 2 :

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

$\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{c}7\\ 11\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{c}\left(1\right)\left(7\right)+\left(2\right)\left(11\right)\\ \left(3\right)\left(7\right)+\left(4\right)\left(11\right)\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{c}7+22\\ 21+44\end{array}\right]\phantom{\rule{0ex}{0ex}}⇒\left[\begin{array}{c}29\\ 65\end{array}\right]$