For some matrices \((A, B)\), it

• does not commute, so

  \(A B \neq B A\)

• does not cancel, so

  \(\bcancel{A} B = \bcancel{A} C \,\,\,\,\, \not \!\!\!\!\! \iff B = C\)

• does not zero-multiply, so

  \(A B = 0 \,\,\,\,\, \not \!\!\!\!\! \iff A = 0 \lor B = 0\).