Question

the transformation $t = \

$$\begin{bmatrix} 0 & -1 \\\\ -1 & 0 \\end{bmatrix}$$

$ is applied the figure below
(image of a coordinate plane with triangle abc: a at (1,4), b at (3,4), c at (1,1))
what type of transformation of triangle abc was applied by t?
\bigcirc rotate 180° counterclockwise about the origin
\bigcirc reflect across the line $y = x$
\bigcirc reflect across the line $y = -x$
\bigcirc rotate 270° counterclockwise about the origin

Explanation:

Step1: Recall rotation matrix

The rotation matrix for a rotation of \(\theta\) counter - clockwise about the origin is \(R=

$$\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$$

\).
For a rotation of \(270^{\circ}\) counter - clockwise, \(\theta = 270^{\circ}\), \(\cos(270^{\circ})=0\), \(\sin(270^{\circ})=- 1\). Then \(R=

$$\begin{bmatrix}0&-(-1)\\-1&0\end{bmatrix}$$

=

$$\begin{bmatrix}0&1\\-1&0\end{bmatrix}$$

\).
The reflection matrix about the line \(y = x\) is \(M=

$$\begin{bmatrix}0&1\\1&0\end{bmatrix}$$

\).
The reflection matrix about the line \(y=-x\) is \(N=

$$\begin{bmatrix}0&-1\\-1&0\end{bmatrix}$$

\).
The rotation matrix for a \(180^{\circ}\) counter - clockwise rotation is \(P=

$$\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$$

\).

Answer:

reflect across the line \(y = -x\)