Matrix multiplication

• What Are the Practical Uses of Matrix Multiplication?
  Matrix multiplication combines transformations and efficiently represents calculations in graphics, systems, networks, probabilities, and machine learning.
• If A Commutes with Every Matrix B, Must A Be cI?
  For square matrices over a field, commuting with every matrix forces A to be scalar; see a proof using matrix units and a worked 2×2 example.
• How to Combine Transformation Matrices
  Combine transformation matrices by multiplying them in reverse application order for column vectors, including translations with homogeneous coordinates.
• What Conditions Make the Product of Two Matrices Zero?
  For compatible matrices, AB = 0 exactly when the image of B lies in the kernel of A; this also implies a rank bound.