# Combination of Transformations

# Combination of Transformations

- One fundamental concept of matrix transformations is that they can be
**combined**. - When combining transformations,
**the order of operations**matters, just like with regular numbers. - Combining matrix transformations means
**multiplying the matrices**. Remember, matrix multiplication is not commutative! - Transformations can be combined to achieve more complex effects. For instance, scaling followed by rotation, or rotation followed by translation, etc.
- The
**resultant matrix**of combining transformations carries out the desired transformations in the correct order. - To find the matrix that represents
**two successive transformations**, you need to multiply the matrix of the first action by the matrix of the second. - Given two matrix transformations A and B, the combined transformation BA
**first applies B**, then applies A. - Be aware that the
**combined effect**of the transformation is usually not just the sum of the individual effects. - Matrix multiplication is
**associative**. That is, for any three matrices A, B, and C, (AB)C = A(BC). This is handy when calculating successive transformations.

# Inversion of Transformations

- Every matrix transformation has an
**inverse transformation**. - The inverse of a transformation is a second transformation that
**undoes the effect**of the first. - If A is the matrix of an original transformation, then
**A^-1**, the inverse matrix of A, represents the**inverse transformation**. - Multiplying a matrix A by its inverse A^-1 (in either order) gives the
**identity matrix**. - The identity matrix
**doesn’t change**elements of the space.