How I do Matrix Multiplication

I am in the process of learning linear algebra, and have struggled finding a good way to multiply matrices by hand. Today I finally found a process I like. It meets two criteria for me:

Clear: its easy to look back and see your process for deriving the product. It is easy to find mistakes.
Geometric: one can see how the individual vectors are affected via the transformation.

This method may not be the most efficient, but that develops with practice. Right now I think its important I understand what I'm doing.
Lets go through a problem using my method.

$\text{Let }A=\begin{bmatrix}1&2\\3&4\end{bmatrix},B=\begin{bmatrix}-2&4\\-3&1\end{bmatrix}\newline H=AB\newline\text{Find }H$

Lets find the value of the first column of H.

$H=\begin{bmatrix}1&2\\3&4\end{bmatrix} \begin{bmatrix}-2&4\\-3&1\end{bmatrix}\newline\newline \text{column 1: } \begin{bmatrix}1&2\\3&4\end{bmatrix} \begin{bmatrix}-2\\-3\end{bmatrix}=-2 \begin{bmatrix}1\\3\end{bmatrix} + (-3)\begin{bmatrix}2\\4\end{bmatrix} = \begin{bmatrix}-2\\-6\end{bmatrix} + \begin{bmatrix}-6\\-12\end{bmatrix} = \begin{bmatrix}-8\\-18\end{bmatrix}$

Next lets find the second column.

$\text{column 2: } \begin{bmatrix}1&2\\3&4\end{bmatrix} \begin{bmatrix}4\\1\end{bmatrix}=4 \begin{bmatrix}1\\3\end{bmatrix} + 1\begin{bmatrix}2\\4\end{bmatrix} = \begin{bmatrix}4\\12\end{bmatrix} + \begin{bmatrix}2\\4\end{bmatrix} = \begin{bmatrix}6\\16\end{bmatrix}$

Sweet. Now we just need to put both columns back together and we have H.

$H=\begin{bmatrix}-8&6\\-18&16\end{bmatrix}$

This is how I do matrix multiplication on paper. I try not to skip steps so I do not make a mistake. Here's the whole process symbolically:

$M=\begin{bmatrix}w&y\\x&z\end{bmatrix}\begin{bmatrix}a&c\\b&d\end{bmatrix}\newline\newline \text{column 1: }\begin{bmatrix}w&y\\x&z\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}=a\begin{bmatrix}w\\x\end{bmatrix}+b\begin{bmatrix}y\\z\end{bmatrix}=\begin{bmatrix}aw\\ax\end{bmatrix}+\begin{bmatrix}by\\bz\end{bmatrix}=\begin{bmatrix}aw+by\\ax+bz\end{bmatrix}\newline \text{column 2: } \begin{bmatrix}w&y\\x&z\end{bmatrix}\begin{bmatrix}c\\d\end{bmatrix}=c\begin{bmatrix}w\\x\end{bmatrix}+d\begin{bmatrix}y\\z\end{bmatrix}=\begin{bmatrix}cw\\cx\end{bmatrix}+\begin{bmatrix}dy\\dz\end{bmatrix}=\begin{bmatrix}cw+dy\\cx+dz\end{bmatrix}\newline\newline \therefore\newline\newline M=\begin{bmatrix}aw+by&cw+dy\\ax+bz&cx+dz\end{bmatrix} \newline\newline\blacksquare$

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