## Decomposing Transformations

Matrices representing transformations may be multiplied by each other, representing a composition of transformations. We also need to be able to express a matrix as a product of matrices representing transformations. The transformations are are particularly interested in are enlargements or scalings, rotations and reflections.

Suppose then that we have the matrix

This matrix has positive determinant so one of the transformations involved is a rotation.

The determinant of the matrix isWe can divide each element of the matrix byto give

The determinant of this matrix is 1 and the each column and row is of magnitude one so the matrix is a rotation matrix. We can equate this to the rotation matrix and find the angle of rotation.

Identifying matrix entries in the upper left and lower left positions gives the equations

ThenThe matrix represents an enlarge with scale factorcentre the origin and a rotation anticlockwise by 69.44 degrees.

The order of transformation in this particular case has no effect because the matrix representing the enlargement isand this is a multiple of the identity.