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?
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.