Question

Compute the products$\mathit{A}\overrightarrow{\mathbf{x}}$ in Exercises 16 through 19 using paper and pencil (if the products are defined).

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

Short answer

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

Step-by-step solution

Consider the given matrices 

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

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

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 $\overrightarrow{x}$is$\left[\begin{array}{c}1\\ 2\end{array}\right]$ .

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 $\left[\begin{array}{c}5\\ 11\\ 17\end{array}\right]$.