Marc Swadener is Assistant Professor of Mathematics Education at the University of Colorado, Boulder, and also Assistant Director of the Undergraduate PreService Teacher Education Program there, which is funded by a NSF grant. He is editor of the Colorado Mathematics Teacher, has written articles for the A rithmetic Teacher, and is active in state and national organizations. Many articles in journals and textbooks deal with the utilization of matrices and matrix operations to characterize transformations on the plane. However, most of these deal only with special cases of the four basic transformations on the plane: translations, rotations, reflections, and dilations (sometimes called expansions and contractions, or magnifications). The special cases considered are rotations of 00 about the origin, reflections about the x and y axes, and dilations about the origin. Knowledge of these special cases and the general case of translations (usually considered) allows us, with relative ease, to establish a procedure for obtaining the matrices which characterize the general case of these transformations on the plane, and ultimately obtain the matrices themselves. The general cases are rotation of 00 about any point (a, b), reflection about any line y = mx + b, and dilation about any point (a, b) with dilation factor k. The approach taken will be to use 3 x 3 matrices as the characterizing matrices, and to use 3 x 1 homogeneous column matrices as the variable matrices in the matrix expression used to find the image of a point on the plane under a transformation. The operation used will be matrix multiplication. That is, if M is the 3 x 3 matrix characterizing a given transformation, and Cis the 3 x 1 homogeneous column matrix corresponding to a point P on the plane, then M * C= C' is the 3 x 1 homogeneous column matrix corresponding to the image of P under the transformation. It should be pointed out that if the coordinates of a point P are (x,y), then coordinates of the form (x, y, k) are called homogeneous coordinates for point P, and