Answer:
[tex]T = \left[\begin{array}{ccc}-\frac{1}{\sqrt{2} } &\frac{1}{\sqrt{2} }\\\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} }\end{array}\right][/tex]
Step-by-step explanation:
Let General Transformation matrix be denoted as T
Step 1: Clockwise rotation of 45 degrees
General counterclockwise rotation matrix in 2-dimension is given as
[tex]R(\theta)=\left[\begin{array}{ccc}cos\theta & - sin\theta\\sin\theta&cos\theta\\\end{array}\right][/tex]
For clockwise rotation we need to insert θ as negative in the above matrix. Therefore, the resulting matrix is
[tex]R(-\theta)=\left[\begin{array}{ccc}cos\theta & sin\theta\\-sin\theta&cos\theta\\\end{array}\right][/tex]
as sin(-θ) = -sin (θ) and cos(-θ) = cos (θ)
For 45 degrees
[tex]sin(45) = \frac{1}{\sqrt{2} }[/tex] and [tex]cos(45) = \frac{1}{\sqrt{2} }[/tex]
[tex]R(-45)=\left[\begin{array}{ccc}\frac{1}{\sqrt{2} } & \frac{1}{\sqrt{2} }\\-\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} }\\\end{array}\right][/tex]
Step 2: Reflection through line y = x
This type of reflection maps (x,y)→(y,x)
Therefore the general matrix is
[tex]R(x,y)=\left[\begin{array}{ccc}0&1\\1&0\end{array}\right][/tex]
Step 3: General Transformation Matrix
T = R(x,y) R(-θ)
[tex]T=\left[\begin{array}{ccc}0&1\\1&0\end{array}\right] \left[\begin{array}{ccc}\frac{1}{\sqrt{2} } & \frac{1}{\sqrt{2} }\\-\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} }\\\end{array}\right][/tex]
[tex]T = \left[\begin{array}{ccc}-\frac{1}{\sqrt{2} } &\frac{1}{\sqrt{2} }\\\frac{1}{\sqrt{2} }&\frac{1}{\sqrt{2} }\end{array}\right][/tex]
