## Problem on matrix multiplication

Find the $3 \times 3$ matrix $\bfA$ satisfying \begin{align}
\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
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 $$\bfA = \begin{pmatrix}1 & 4 & 7\\2&5&8\\3&6&9 \end{pmatrix}.$$