Inverse matrix of transformation matrix (rotation and translation matrix)

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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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