Quick questions

Given an arbitrary 3x3 matrix, how do you know it's a homography?

Given an arbitrary 3x3 matrix, how do you know it's an affine transformation?

Given an arbitrary 3x3 matrix, how do you know it's an Euclidean transformation?

Answers:

1) Should be invertible.

2) Should be invertible and last row should be (0,0,1).

3) Should be invertible and last row should be (0,0,1) and upper-left 2x2 matrix should be orthogonal.

How do you know it's invertible?

How do you know the upper-left 2x2 matrix is orthogonal?

Answers

1) determinant should be zero.

2) That means its columns ( & hence rows too) should be orthonormal. Check their dot product.