I was reading: https://en.wikipedia.org/wiki/Metric_tensor#Arclength
Where in it gives the euclidean measure of distance as
$$ ds^2 = E du^2 + 2 F du dv + G dv^2 $$
Equivalently as
$$ ds^2 =\begin{bmatrix} du & dv \end{bmatrix} \begin{bmatrix} E & F \\ F & G\end{bmatrix} \begin{bmatrix} du \\ dv \end{bmatrix} $$
But this blatantly false, considering that
$$ \begin{bmatrix} E & F \\ F & G\end{bmatrix} \begin{bmatrix} du \\ dv \end{bmatrix} = \begin{bmatrix} E du + F dv \\ F du + G dv \end{bmatrix} $$
So then
$$ ds^2 = \begin{bmatrix} du & dv \end{bmatrix} \begin{bmatrix} E du + F dv \\ F du + G dv \end{bmatrix} = $$
$$E du^2 + (E+F) du dv + F dv^2 +F du^2 + (F + G) du dv + G dv^2 =$$
$$(E+F) du^2 + (E + 2F + G) du dv +(F+G) dv^2 $$
And doesn't equate to the original at all. So what exactly are they trying to convey in this section? Obviously it's not classical matrix multiplication.