Matrices follow all the usual laws ofalgebra except commutativity ofmultiplication, meaning that for two matrices A and B it is not generally true that AB=BA. However, matrix multiplication is associative, meaning that products ofany length may be written unambiguously without the need for bracketing.
Linear transformations of the plane are typically rotations about the origin, reflections in lines through the origin, enlargments and contractions about the origin, and so called shears(or slanting),which move points parallel to a fixed axis by an amount proportional to their distance from that axis in a manner similar to the way the pages ofa book can slide past one another. Any sequence of these transformations can be effected by multiplying all of the relevant matrices together to reveal a single matrix that has the same net effect as all those transformations acting in turn.The rows of the resultant matrix are simply the images of the two points(1,0)and(0,1), as we saw above, known as basis vectors.