Determinant (Vol Scale):
1.000
Trace (Sum Diag):
3.000
Kernel Dimension (Nullspace):
0 (Full Rank)
The three dimensional shear preset maps x, y, z to x plus half y, y plus half z, and z. In the z equals zero slice, a unit square becomes a parallelogram: the top edge moves half a unit to the right while the bottom stays put. Its area stays one. The full preset matrix is upper triangular with diagonal ones, so determinant is one and the three dimensional volume multiplier also remains one. The XY Mirror preset keeps x and y and changes z to negative z. The unit vector in the positive z direction becomes the negative z unit vector. Its determinant is negative one. The sign records reversed orientation, while the absolute determinant remains one, so the volume magnitude is unchanged. This distinguishes a mirror from a collapse. The Plane Projection preset maps x, y, z to x, y, zero. Every point with the same x and y lands at the same point regardless of z. Its determinant is zero: the three dimensional volume collapses into a plane. The z direction is a null direction because its unit vector maps to zero. The source’s general kernel display uses a determinant and trace heuristic; it is not a complete rank computation for every arbitrary matrix.