Matrix Force Method of structural analysis

The following is a excerpt from a book which was published by the GK Publishers for the preparation of the GATE exam 2013. I have already written the introductory article to the matrix method, and you can get the understanding of it from my previous article of the same title. In this post I have taken the liberty to post the small content from the book, which will act as the advertisement of the book, so I hope everybody is happy with it.
Matrix method is the method of analysis of the indeterminate structures which uses the matrix approach to get the final moments at various joints in the structure.

Matrix Force and displacement Method: 
In case of analysis of beams by displacement method, joint rotations and displacements due to external forces are assumed as unknowns. Consider a continuous beam ABCD shown in figure(a) and (b) below:

The degree of the, NP, of the continuous beam is 3 that is three numbers of rotations can happen at the three joints B, C and D. The moments P1, P2 and P3 are the unbalanced fixed end moments due to external loads. X1, X2 and X3 are the rotations taken as unknowns.
At each joint, external forces P1, P2 and P3 are in equilibrium with the internal forces F1, F2, F3... F6.  Internal end rotations 'e' are the slopes of the tangents drawn to the elastic curve. Rotations are positive if they are clockwise and they are negative if the rotations are anticlockwise.  Similarly moments P1, P2 and P3 are positive if they are clockwise in direction and negative if anti clock wise.

Static Matrix A: Equilibrium condition between the external forces and the internal forces produces a static matrix.

Refer to the equilibrium diagram shown above,

P1= F2+F3,  P2 = F4+ F5  and P3= F6

                       In Matrix notations:

Here [A] is called a static matrix.

 Deformation Matrix B:

External joint forces P1, P2 and P3 cause joint rotations X1, X2 and X3 etc., these in turn cause internal joint rotations. Refer figure (b), where an external rotation X1= +1 will result in internal rotation e2=e3 = +1;                                                                                                                                                        In Matrix form :                                        Here [B] is called the deformation matrix.

Member stiffness Matrix:
Consider the figure given below:

Positive forces Fi and Fj are shown with the positive rotations ei and ej. Using the principle of conjugate beam it can be proved that,              
Fi = (4EI/l).ei  + (2EI/l).ej
Fj = (2EI/l). ei + (4EI/l). ej
Member stiffness matric 'S' shows the relationship between internal joint moments F1, F2 etc. in terms of joint rotations e1, e2 etc for the span AB, BC and CD.

Now , consider the following equation in the figure below:                                                                 4th equation is followed from 2nd and 3rd equation.
5th and 6th equation is followed from the 1st and 5th equation, first equation being the top most equation.
So the joint displacements X1, X2 are evaluated first from above equation and then joint forces F1, F2 are evaluated.
Final end moments are
F1f= Fixed End Moment1 +  F1
and F2f = Fixed End Moment2 + F2.

So this way you have found the value of the final end moments at the various joints.