The article is devoted to the improvement of the mathematical apparatus used to solve the problems of synthesis and analysis of the kinematics of SCARA industrial robot manipulators. As this device, it is proposed to use the generalized matrices of transformation of coordinates of industrial manipulators. The purpose and relevance are formed on the basis of the presented analysis of the state of research on the topic. The article provides a structural synthesis of structural matrices, where their elements are recorded according to elementary transformations when moving from one coordinate system to another. The formed numerical matrices make it possible to determine the position of any movement point at any moment of time of an arbitrary industrial manipulator according to a given program of movement in variable generalized coordinates. For further calculation of matrices, it is suggested to use Mathcad and Maple computer mathematics programs. The article shows that the orientation of the corresponding link of the manipulator in the working space of the robot depends on the structural construction of the manipulator, namely on the corresponding sizes of the links and kinematic pairs located on the path of passage along the kinematic sequence from the stationary link to the considered one. The synthesis of industrial robot movement trajectories and the orientation of its links, which can be obtained using the formed coordinate transformation matrices, make it possible to compare functional trajectories for the purpose of their further optimization and improvement.