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Q18E

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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 ${\mathbit{A}}\stackrel{\mathbf{\to }}{\mathbf{x}}$ in Exercises 16 through 19 using paper and pencil (if the products are defined).18. $\mathbf{\left[}\begin{array}{cc}\mathbf{1}& \mathbf{2}\\ \mathbf{3}& \mathbf{4}\\ \mathbf{5}& \mathbf{6}\end{array}\mathbf{\right]}\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{2}\end{array}\mathbf{\right]}$

The product of$A\stackrel{\to }{x}$ is, $\left[\begin{array}{c}5\\ 11\\ 17\end{array}\right]$ .

See the step by step solution

## Step 1: Consider the given matrices

If ${\mathbf{A}}$ is an${\mathbf{n}}{\mathbf{×}}{\mathbf{m}}$ matrix with row vectors${\stackrel{\mathbf{\to }}{\mathbf{\omega }}}_{{\mathbf{1}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{....}}{\mathbf{,}}{\stackrel{\mathbf{\to }}{\mathbf{\omega }}}_{{\mathbf{n}}}$ and $\stackrel{\mathbf{\to }}{\mathbf{x}}$is a vector in ${{ℝ}}^{{m}}$ then, .

${\mathbit{A}}\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{=}}\mathbf{\left[}\begin{array}{c}\mathbf{-}{\stackrel{\mathbf{\to }}{\mathbf{\omega }}}_{\mathbf{1}}\mathbf{-}\\ \mathbf{⋮}\\ \mathbf{-}{\stackrel{\mathbf{\to }}{\mathbf{\omega }}}_{\mathbf{n}}\mathbf{-}\end{array}\mathbf{\right]}\stackrel{\mathbf{\to }}{\mathbf{x}}{\mathbf{=}}\mathbf{\left[}\begin{array}{c}\mathbf{-}{\stackrel{\mathbf{\to }}{\mathbf{\omega }}}_{\mathbf{1}}\mathbf{\cdot }\stackrel{\mathbf{\to }}{\mathbf{x}}\mathbf{-}\\ \mathbf{⋮}\\ \mathbf{-}{\stackrel{\mathbf{\to }}{\mathbf{\omega }}}_{\mathbf{n}}\mathbf{\cdot }\stackrel{\mathbf{\to }}{\mathbf{x}}\mathbf{-}\end{array}\mathbf{\right]}$

The given expression is,

$\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]$

Here$A$ is $\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]$and $\stackrel{\to }{x}$ is$\left[\begin{array}{c}1\\ 2\end{array}\right]$ .

## Step 2: Perform the operation

Now, find the product by performing the row column operations as follow:

$\begin{array}{c}\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]=\left[\begin{array}{c}1\cdot 1+2\cdot 2\\ 3\cdot 1+4\cdot 2\\ 5\cdot 1+6\cdot 2\end{array}\right]\\ =\left[\begin{array}{c}5\\ 11\\ 17\end{array}\right]\end{array}$

Hence, the required product is $\left[\begin{array}{c}5\\ 11\\ 17\end{array}\right]$ .

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