For all matrices \(A, B, C\), vectors \(\vec{u}, \vec{v}\), and scalars \(c\), it

• associates, so

  \((A B) C = A (B C)\),

• associates with respect to matrix scaling and vector scaling, so

  \(c ( A B ) = ( c A ) B = A ( c B )\) \(\qand\) \(c ( A \vec{u} ) = ( c A ) \vec{u} = A ( c \vec{u} )\),

• left-distributes over matrix addition and vector addition, so

  \(A ( B + C ) = A B + A C\) \(\qand\) \(A ( \vec{u} + \vec{v} ) = A \vec{u} + A \vec{v}\)

• and its multiplicative identity is the identity matrix, so

  \(I_m A = A I_n = A\) \(\qand\) \(I_n \vec{u} = \vec{u}\).