 set of alternatives the players choose.[57] Text mining and automated thesaurus compilation makes use of document-term matrices such as tf-idf to track frequencies of certain words in several documents.[58]Complex numbers can be represented by particular real 2-by-2 matrices viaa+ib↔[a−bba],{\displaystyle a+ib\leftrightarrow {\begin{bmatrix}a&-b\\b&a\end{bmatrix}},}under which addition and multiplication of complex numbers and matrices correspond to each other. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above. A similar interpretation is possible for quaternions.[59]Early encryption techniques such as the Hill cipher also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break.[60] Computer graphics uses matrices both to represent objects and to calculate transformations of objects using affine rotation matrices to accom