Can you prove why Popsicle Stick Multiplication works?

Question

This is a unique way of multiplying numbers by using sticks. Let's call it "Popsicle Stick Multiplication". Or maybe "Linear Algebra" quite literally. Take a look at both images that I've drawn below. Can someone explain why this works out? Notice how you must carry the 10's digit when the sum of the intersections is greater than 9, and if you have more than 99, then carry the hundreds digit just like any other multiplication problem. I'm not sure if it's faster to do it the long way or to do it this way. Perhaps this is the way the Egyptians did it while building the great pyramids? This works for multiplying any amount of sticks.

Answer 1

$$\begin{matrix} & 1 & 2 \\ \times & 3 & 4 \\ \hline & 4 & 8 \\ 3 & 6 \\ \hline 3 & 10 & 8 \\ \hline 4 & 0 & 8 \end{matrix}$$

If you tilt your head a little it's exactly like popsicle stick multiplication. You're just doing the carries later rather than sooner, compared to the usual way to do long multiplication.

Update:

$$\begin{matrix} & & & 4 & 1 & 2 \\ \times & & 1 & 2 & 3 & 1 \\ \hline & & & 4 & 1 & 2 \\ & & 12 & 3 & 6 \\ & 8 & 2 & 4 \\ 4 & 1 & 2 \\ \hline 4 & 9 & 16 & 11 & 7 & 2 \\ \hline 5 & 0 & 7 & 1 & 7 & 2 \end{matrix}$$

Answer 2

Consider the first image: $$12*34 = (10 + 2)(30 + 4) = (1*3)*10^2 + (2*3 + 1*4)*10^1 + (2*4)*10^0$$ This is just a visual form of the distributive law.