Skip to main content\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}}
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\)
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}
then
\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}
Checkpoint 3.1. Try it yourself: Scalar Multiplication.
Compute:
\begin{equation}
i\cdot \begin{pmatrix}
1\amp i\\
-2i\amp 3
\end{pmatrix}\text{.}\tag{3.2.4}
\end{equation}
Solution.
\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}
