Q44. - Algebra 2

Question

Which matrix represents a reflection over the y-axis followed by a reflection over the x-axis?

 A $\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]$      B $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$    C $\left[\begin{array}{cc}-1& -1\\ -1& -1\end{array}\right]$    D none of these

Step-by-Step Solution

Answer

The matrix represents a reflection over the y-axis followed by a reflection over the x-axis is $\text{B }\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

Step 1 - Vertex matrix

Let $\left(x,y\right)$ be an ordered pair then its vertex matrix will be $\left[\begin{array}{c}x\\ y\end{array}\right]$

Step 2 - Vertex matrix after reflection over the y - a x i s

Vertex matrix of new point $\left(x_1,y_1\right)$ after reflection over the y -axis is product matrix $\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$ with vertex matrix of original point as shown below

$\left[\begin{array}{c}{x}_1\\ y_1\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\times \left[\begin{array}{c}x\\ y\end{array}\right]$

Step 3 - Vertex matrix after reflection over the x - a x i s

Vertex matrix of new point $\left(x_2,y_2\right)$ after reflection over the x -axis is product matrix $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ with vertex matrix of point $\left(x_1,y_1\right)$ and use the associative property of matrix multiplication as shown below

 $\begin{array}{c}\left[\begin{array}{c}{x}_2\\ y_2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\times \left[\begin{array}{c}{x}_1\\ y_1\end{array}\right]\\ =\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\times \left(\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\times \left[\begin{array}{c}x\\ y\end{array}\right]\right)\\ =\left(\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\times \left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\right)\times \left[\begin{array}{c}x\\ y\end{array}\right]\\ =\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]\times \left[\begin{array}{c}x\\ y\end{array}\right]\end{array}$

Thus, the matrix represents a reflection over the y-axis followed by a reflection over the x-axis is $\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$