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.