What the matrix changes

A 2D transformation multiplies a vector by a matrix AA to create a new vector mathbfv′mathbf{v'}.

Different matrices rotate, stretch, reflect, or shear the plane. The determinant tells you whether area is preserved, enlarged, flipped, or collapsed.

Use simple test vectors first, then change one matrix entry at a time to see which axis it affects.

Quick checks

  • An identity matrix leaves the vector unchanged.
  • A zero determinant means the plane is flattened into lower dimension output.
  • Swapping columns or signs often reveals reflections and rotations faster than reading the numbers alone.

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