Multiplication with a scalar

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A matrix is multiplied with a scalar by multiplying all the components with the scalar. The result is again a matrix with the same dimensions:

$\mathbf{A} \cdot c =c \cdot \mathbf{A}=c\cdot\left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{array}\right]= \left[\begin{array}{ccc} c\cdot a_{11} & c\cdot a_{12} & c\cdot a_{13}\\ c\cdot a_{21} & c\cdot a_{22} & c\cdot a_{23}\\ c\cdot a_{31} & c\cdot a_{32} & c\cdot a_{33} \end{array}\right]$

Example: Multiplying a matrix with a scalar $\begin{align} \mathbf{A}&=\left[ \begin{array}{ccc} 2 & 3 & 0\\ 4 & 1 & 2\\ 3 & 1 & 4 \end{array}\right],\quad c=3\\ c \cdot\mathbf{A}&=3\cdot\left[\begin{array}{ccc} 2 & 3 & 0\\ 4 & 1 & 2\\ 3 & 1 & 4 \end{array}\right]= \left[\begin{array}{ccc} 3\cdot 2 & 3\cdot 3 & 3\cdot 0\\ 3\cdot 4 & 3\cdot 1 & 3\cdot 2\\ 3\cdot 3 & 3\cdot 1 & 3\cdot 4 \end{array}\right]= \left[\begin{array}{ccc} 6 & 9 & 0\\ 12 & 3 & 6\\ 9 & 3 & 12 \end{array}\right] \end{align}$