Inverse matrix of transformation matrix (rotation and translation matrix)


4x4 matrix [R|t] is the mixture of 3x3 rotation matrix R and translation 3D vector t. Let's call [R|t] transformation matrix.

Lemma
Multiplying rotation matrix and rotation matrix results in rotation matrix
Lemma
Multiplying transformation matrix and transformation matrix results in transformation matrix
Lemma
The inverse of rotation matrix is its transpose
Lemma
The inverse of transformation matrix always exists
Proof
It is apparant from the definition of rotation matrix and transformation matrix

The inverse of transformation matrix [R|t] is [R^T | - R^T t].


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