The matrix shown is a 3x3 matrix plus a translation row (the three t's on the bottom row).
If you have a 4x4 matrix instead, you can still use this method for the first 3 columns of the 4x4 matrix (the fourth column in 4x4 matrices is the scaling information - so this method used on 4x4 matrices will do the translation and rotation but not the scaling).
Systems that use 3x3 matrices will often implement scale by scaling the trigonomic values in each matrix position (so instead of a value from -1 to 1, at half scale it would be -0.5 to 0.5)... thus this method in a 3x3 matrix system will usually scale your vector too.