Problem on matrix multiplication

$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
Find the $3 \times 3$ matrix $\bfA$ satisfying \begin{align}
\bfA \begin{pmatrix}1\\0\\0\end{pmatrix} &= \begin{pmatrix}1\\2\\3\end{pmatrix}, \quad
\bfA \begin{pmatrix}0\\1\\0\end{pmatrix} = \begin{pmatrix}4\\5\\6\end{pmatrix}, \quad
\bfA \begin{pmatrix}0\\0\\1\end{pmatrix} = \begin{pmatrix}7\\8\\9\end{pmatrix}.\\
\end{align}

  • Solution

    To find $\bfA$, let's give a name to each of its unknown components. Then, we can carry out the matrix multiplications and analyze what we get.
    Let $\bfA = \begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}.$ We are trying to find $a, b, c, \ldots$
    We now write out the matrix multiplications given in the problem statement.
    We would like to express $\bfA \begin{pmatrix}1\\0\\0 \end{pmatrix}$ in terms of $a, b, c, \ldots$
    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 $\mathbf{B}$.
    Because $\bfA$ is $3 \times 3$ and is multiplied by a $3 \times 1$ vectors, the result is $3 \times 1$.
    We can compute that $$ \begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix} \begin{pmatrix}1\\0\\0\end{pmatrix} = \begin{pmatrix}a\\d\\g\end{pmatrix}.$$
    The problem statement tells us that this matrix product is $\begin{pmatrix}1\\2\\3\end{pmatrix}$.
    Hence, $a = 1$, $d=2$, $g =3$.
    Similarly, we can compute that $$ \begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix} \begin{pmatrix}0\\1\\0\end{pmatrix} = \begin{pmatrix}b\\e\\h\end{pmatrix} = \begin{pmatrix}4\\5\\6\end{pmatrix}$$
    Hence, $b =4$, $e=5$, $h = 6$.
    Finally, we can compute that $$ \begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix} \begin{pmatrix}0\\1\\0\end{pmatrix} = \begin{pmatrix}c\\f\\i\end{pmatrix} = \begin{pmatrix}7\\8\\9\end{pmatrix}$$
    Hence, $c=7$, $f=8$, $i=9$.
    We conclude that $$\bfA = \begin{pmatrix}1 & 4 & 7\\2&5&8\\3&6&9 \end{pmatrix}.$$

Related topics