Chapter 4.4, Problem 44HP

(a)

To determine

To find: The transformation matrix for the reflection of $y=-x$ .

Answer to Problem 44HP

The matrix $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ after reflection is $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ .

Explanation of Solution

Given:

The given equation is $y=-x$ .

Calculation:

Consider the matrix is,

  $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

Transform the matrix across $y=-x$ as,

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

(b)

To determine

To find: The transformation matrix for the rotation $90°$ in the clockwise direction.

Answer to Problem 44HP

The transformed matrix is $\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ .

Explanation of Solution

Given:

The transformation is done as rotation $90°$ in the clockwise direction.

Calculation:

Consider the unit vectors as,

  $\left[\begin{array}{c}1\\ 0\end{array}\right]$ and $\left[\begin{array}{c}0\\ 1\end{array}\right]$

The clockwise rotation of the matrix in the $90°$ in the clockwise direction.

  $\left[\begin{array}{c}0\\ 1\end{array}\right]$ and $\left[\begin{array}{c}0\\ -1\end{array}\right]$

Thus, the transformation matrix is,

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