# 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.