Why there is a factor 2/3 added to Clarke Transform matrix?
If you do the transform without the 2/3 scale factor, the amplitude of the alpha-beta variables is 1.5 times higher than that of the ABC variables.  The scaling is done only to maintain the amplitude across the transform.  For example, taking a balanced 3-phase system having amplitude 1, the first row becomes
$$
\cos{\omega t}-\frac{1}{2}\cos{(\omega t+\frac{2\pi}{3})}-\frac{1}{2}\cos{(\omega t-\frac{2\pi}{3})}
$$
Using the identity 
$$
\cos{A}\cos{B} = \frac{1}{2}\cos{(A+B)} + \frac{1}{2}\cos{(A-B)}
$$
this is reduced to 
$$
\cos{\omega t} - \cos{(\frac{2\pi}{3}})\cos{\omega t} = \frac{3}{2}\cos{\omega t}
$$