Scalar multiplication

The scalar multiplication of a matrix A = (aij) and a scalar r gives a product r A of the same size as A. The entries of r A are given by

 (r\mathbf{A})_{ij} = r \cdot a_{ij}. \,

For example, if

\mathbf{A}=\begin{bmatrix} a & b \\ c & d \end{bmatrix}

then

 r \cdot \mathbf{A}=\begin{bmatrix} r \cdot a & r \cdot b \\ r \cdot c & r \cdot d \end{bmatrix}.

If we are concerned with matrices over a more general ring, then the above multiplication is the left multiplication of the matrix A with scalar r while the right multiplication is defined to be

 (\mathbf{A}r)_{ij} = a_{ij} \cdot r. \,

When the underlying ring is commutative, for example, the real or complex number field, the two multiplications are the same. However, if the ring is not commutative, such as the quaternions, they may be different. For example


  i\begin{bmatrix} 
    i & 0 \\ 
    0 & j \\ 
  \end{bmatrix}
= \begin{bmatrix}
    -1 & 0 \\
     0 & k \\
  \end{bmatrix}
\ne \begin{bmatrix}
    -1 & 0 \\
    0 & -k \\
  \end{bmatrix}
= \begin{bmatrix}
    i & 0 \\
    0 & j \\
  \end{bmatrix}i.