Having to do copious calculations by hand when preparing for an exam, I came to realize that there was an alternative way of interpreting a matrix multiplication. This new insight would allow me to instantly guess the following product *without ever doing any numerical multiplication*:

\[\begin{bmatrix}

1 & 2 & 3 \\

4 & 5 & 6 \\

7 & -8 & 0

\end{bmatrix}

\begin{bmatrix}

0 & 0 & 0 \\

0& 1 & 0 \\

1& 0 & 0

\end{bmatrix}

=

\begin{bmatrix}

3 & 2 & 0 \\

6 & 5 & 0 \\

0 & -8 & 0

\end{bmatrix}\]

Was there a way to have known that the first column of the product would be the third column of the first matrix?

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