Question

For exercise 29-32, use rectangle $ABCD$ with vertices $A\left(-4,4\right),B\left(4,4\right),C\left(4,-4\right),\text{and}D\left(-4,-4\right)$

31. Find the coordinate of the image in the matrix form after a reflection over the line $y=x$.

Short answer

Coordinate of the image in the matrix form after a reflection over the x-axis followed by reflection over the y-axis

$\left[\begin{array}{cccc}4& -4& -4& 4\\ -4& -4& 4& 4\\ & & & \end{array}\right]$

Step-by-step solution

- Vertex matrix 

An ordered pair $\left(x,y\right)$is represented by a column matrix $\left[\begin{array}{c}x\\ y\\ \end{array}\right]$which is known as vertex matrix of an ordered pair

- Vertex matrix of given rectangle

Since, a rectangle has 4 ordered pair on vertices so required matrix has 2 rows and 4 columns.

First row represent $x-\text{coordinate}$ and second row represents $y-\text{coordinate}$ of each vertex.

As coordinates of the vertices of rectangle are $A\left(-4,4\right),B\left(4,4\right),C\left(4,-4\right),\text{and}D\left(-4,-4\right)$

Write each ordered pair as a column of a matrix to get

$\left[\begin{array}{cccc}-4& 4& 4& -4\\ 4& 4& -4& -4\\ & & & \end{array}\right]$

- Vertex matrix after reflection over the line y=x

Vertex matrix of new rectangle after reflection over the line $y=x$is product of matrix $\left[\begin{array}{cc}0& 1\\ 1& 0\\ & \end{array}\right]$with vertex matrix of given rectangle as shown below

$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\times \left[\begin{array}{cccc}-4& 4& 4& -4\\ 4& 4& -4& -4\end{array}\right]=\left[\begin{array}{cccc}4& 4& -4& -4\\ -4& 4& 4& -4\end{array}\right]$