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Section 3.2 Scalar Multiplication

A matrix can be multiplied by a scalar, in which case each element of the matrix is multiplied by the scalar. In components,

\begin{equation} C_{ij}=\lambda A_{ij}\tag{3.2.1} \end{equation}

where \(\lambda\) is a scalar, that is, a complex number. For example, if

\begin{equation} A = \begin{pmatrix} a\amp b\\ c\amp d \end{pmatrix}\text{,}\tag{3.2.2} \end{equation}


\begin{equation} 3A=3\cdot \begin{pmatrix} a\amp b\\ c\amp d \end{pmatrix} = \begin{pmatrix} 3a\amp 3b\\ 3c\amp 3d \end{pmatrix}\text{.}\tag{3.2.3} \end{equation}


\begin{equation} i\cdot \begin{pmatrix} 1\amp i\\ -2i\amp 3 \end{pmatrix}\text{.}\tag{3.2.4} \end{equation}
\begin{align} i \cdot \begin{pmatrix}1\amp i\\ -2i\amp 3\end{pmatrix} \amp= \begin{pmatrix}(i)(1)\amp (i)(i)\\ (i)(-2i)\amp (i)(3)\end{pmatrix}\notag\\ \amp= \begin{pmatrix}i\amp -1\\ 2\amp 3i\end{pmatrix}\text{.}\tag{3.2.5} \end{align}