Problem on matrix multiplication

$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
Compute the matrix multiplications $$\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}\begin{pmatrix} 1 \\2\\3\end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 1 \\2\\3\end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \end{pmatrix}.$$

  • Solution

    Recall that
    The matrix product of an $n \times m$ matrix with an $m \times \ell$ matrix is an $n \times \ell$ matrix.

    The $(i,j)$ entry of the matrix product $\bfA \mathbf{B}$ is the dot product of the $i$th row of $\bfA$ with the $j$th column of $\mathbf{B}$.
    The first matrix product is a $1 \times 3$ matrix multiplied by a $3 \times 1$ matrix. The result will be a $1\times 1$ matrix. $$\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}\begin{pmatrix} 1 \\2\\3\end{pmatrix} = 1 \cdot 1 + 2 \cdot 2 + 3 \cdot 3 = 14.$$
    The second matrix product is a $3 \times 1$ matrix multiplied by a $1 \times 3$ matrix. The result will be a $3\times 3$ matrix.
    The $(i,j)$ entry of $\begin{pmatrix} 1 \\2\\3\end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \end{pmatrix}$ the $i$th row of $\begin{pmatrix}1\\2\\3\end{pmatrix}$ multiplied by the $j$th column of $\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}.$
    For example, the first row of $\begin{pmatrix} 1 \\2\\3\end{pmatrix}$ is 1, and the second column of $\begin{pmatrix}1&2&3\end{pmatrix}$ is 2. Hence, the $(1,2)$ entry of their product is $1\cdot 2$.
    Computing all nine values, we get $$\begin{pmatrix} 1 \\2\\3\end{pmatrix} \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix}1\cdot 1 & 1 \cdot 2 & 1\cdot 3 \\ 2 \cdot 1 & 2\cdot 2 & 2 \cdot 3 \\ 3 \cdot 1 & 3 \cdot 2 & 3 \cdot 3 \end{pmatrix} = \begin{pmatrix}1 & 2 & 3\\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix}$$

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