Chapter 2.4, Problem 10CFU

To determine

To state: the work rate of the Beatrice who can wallpaper a room in 8 hours.

Answer to Problem 10CFU

  ${\text{A}}^{\text{-1}}\text{=}\left[\begin{array}{cc}\text{1}& \text{-2}\\ \frac{\text{-3}}{\text{2}}& \frac{\text{7}}{\text{2}}\end{array}\right]$ .

Explanation of Solution

Given:

  $\text{wallpaper}\text{}\text{ a room in 8 hours}$ .

Concept used:

The image of linear transformation or matrix is the span of the vector of the linear transformation.

It can be written as $Im\left(A\right)$ .

For example, consider the matrix $A$ which is equal to:

  $\left[\begin{array}{cc}1& 0\\ 0& 2\\ 0& 1\end{array}\right]$

Multiplying this by $2\times 1$ gives a $3\times 1$ matrix. However, regardless of what vector is chosen to multiply by, there are some vectors that can’t be the result.

Result is not image of $A$ . The vector that are possible belong to the span of $A$ . In this case, the span be represented by a parametrized matrix, where $t\text{ and }s$ can be any number:

  $\left[\begin{array}{cc}1& 0\\ 0& 2\\ 0& 1\end{array}\right]\times \left[\begin{array}{c}s\\ t\end{array}\right]=\left[\begin{array}{c}s\\ 2t\\ t\end{array}\right]=Im\left(A\right)$ .

Matrices multiplication:

  $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]=\left[\begin{array}{cc}ae+ag& af+bh\\ ce+dg& cf+dh\end{array}\right].$

Calculation:

The wind was blowing quite strongly when jenny was baby-sitting. she was outside with the children, and they were throwing their large plaster ball up into the air. The wind blew the ball so that it landed approximately $3$ feet east and $4$ feet north of where it was thrown into the car.

If it is to be consider $\text{'}O\text{'}\left(0,0\right)$ is the original location of the ball. The wind blew the ball so, that it landed approximately $3$ feet east and $4$ feet north. That means $3$ feet right and $4$ feet up.

Then $A\left(3,4\right)$ is the translation of the ball it landed at $A$ .

The graph of location of ball is shown below.

The matrix that represents location of translated ball is:

  $\left[\begin{array}{cc}x& 3\\ y& 4\end{array}\right]$ .