Question

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

19.$\mathbf{\left[}\begin{array}{ccc}\mathbf{1}& \mathbf{1}& \mathbf{-}\mathbf{1}\\ \mathbf{-}\mathbf{5}& \mathbf{1}& \mathbf{1}\\ \mathbf{1}& \mathbf{-}\mathbf{5}& \mathbf{3}\end{array}\mathbf{\right]}\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{2}\\ \mathbf{3}\end{array}\mathbf{\right]}$

Short answer

The product $A\overrightarrow{x}$of ,$\left[\begin{array}{ccc}1& 1& -1\\ -5& 1& 1\\ 1& -5& 3\end{array}\right]\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$is.$\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

Step-by-step solution

Consider the given matrices

If $\mathit{A}$is anrole="math" localid="1659684733594" $\mathbf{n}\mathbf{\times }\mathbf{m}$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${\mathit{ℝ}}^{\mathbf{m}}$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,

$\left[\begin{array}{ccc}1& 1& -1\\ -5& 1& 1\\ 1& -5& 3\end{array}\right]\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$

Here$A$ is$\left[\begin{array}{ccc}1& 1& -1\\ -5& 1& 1\\ 1& -5& 3\end{array}\right]$ and $\overrightarrow{x}$is$\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$.

Perform the operation

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

$\begin{array}{c}\left[\begin{array}{ccc}1& 1& -1\\ -5& 1& 1\\ 1& -5& 3\end{array}\right]\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]=\left[\begin{array}{c}1\cdot 1+1\cdot 2+(-1)\cdot 3\\ (-5)\cdot 1+1\cdot 2+1\cdot 3\\ 1\cdot 1+(-5)\cdot 2+3\cdot 3\end{array}\right]\\ =\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\end{array}$

Hence, the required product is $\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$.