# Determine the invertibility of a linear transformation and find the inverse

There is a linear transformation T from R2 to R2 that reflects a vector about the x-axis, then rotate it clockwise by π/3 about the origin and then reflects a vector about the line y = -x.

Determine if the transformation T is invertible or not. If it is invertible, find the standard matrix of the inverse of T.

How could I determine the invertibility of the linear transformation? I calculated the standard matrix of T, and it is 2*2, is it enough to prove T is invertible? Are finding the standard matrix of the inverse of T the same as finding the inverse of the standard matrix of T? I am a little bit confused about the invertibility of a linear transformation. Could someone explain the meaning behind that?

• See the forest among the trees... Trace the steps back: a transformation that first reflects about the line $y=-x$, then rotates anti-clockwise by $\pi/3$, then reflects about the x-axis is obviously the required inverse. Can you proceed from there? – user491874 Dec 12 '17 at 6:33
• Okay. But how can I show the track-back process algebraically? – ninomi Dec 12 '17 at 6:39
• You said you could find the matrix for $T$ - what stops you from doing the same process here, for this transformation I just outlined? – user491874 Dec 12 '17 at 6:42
• So I guess I should do it all again reversely? Thanks! – ninomi Dec 12 '17 at 7:00

Reflection about $x$-axis: $$\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)$$ Clockwise rotation by $\pi/3$: $$\left(\begin{array}{cc}\cos(-\pi/3)&-\sin(-\pi/3)\\\sin(-\pi/3)&\cos(-\pi/3)\end{array}\right)$$ Reflection about the line $y=-x$: $$\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$$ Each of these matrices is invertible. So their product, which will be the matrix $T$, will also be invertible.