Help me intuit matrices

I am reasonably math literate, but I regret to say that the intuition behind matrices is completely lost on me.

Note: Since this forum doesn't appear to support LaTex natively, I recommend using the Tex The World plugin for Firefox, Chrome, or Opera via userscript. When installed, any expression between [; and ;] will render as LaTeX, such as:

[; \sigma(z) \equiv \frac{1}{1+e^{-z}} ;]

1) Let's say I have these matrices, A and B:

[;
A &= \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{pmatrix}
,\quad
B &= \begin{pmatrix}
1 & 4 \\
2 & 5 \\
3 & 6
\end{pmatrix}
;]

What exactly do the values inside of a matrix represent? More specifically, can the values above be graphed, or written in a more familiar notation?

2) Do rows and columns have a particular meaning in relation to matrices? Are all the values in a row logically related to one another?

3) Is the order of values in a row or column of a matrix significant?

4) Wikipedia tells me that, for n x m matrices A and B, matrix addition is defined as:

[;

A + B =
\begin{pmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{pmatrix}

+

\begin{pmatrix}
b_{11} & b_{12} & \cdots & b_{1n} \\
b_{21} & b_{22} & \cdots & b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{m1} & b_{m2} & \cdots & b_{mn} \\
\end{pmatrix}

=

\begin{pmatrix}
a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
\end{pmatrix}
;]

But how is this derived?

6) What does the output of matrix addition represent?

5) Given an n x m matrix A, and an m x p matrix B, matrix multiplication is defined as:

[;
AB &=
\begin{pmatrix}
A_{11} & A_{12} & \cdots & A_{1m} \\
A_{21} & A_{22} & \cdots & A_{2m} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1} & A_{n2} & \cdots & A_{nm} \\
\end{pmatrix}
\begin{pmatrix}
B_{11} & B_{12} & \cdots & B_{1p} \\
B_{21} & B_{22} & \cdots & B_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
B_{m1} & B_{m2} & \cdots & B_{mp} \\
\end{pmatrix}
=
\begin{pmatrix}
(AB)_{11} & (AB)_{12} & \cdots & (AB)_{1p} \\
(AB)_{21} & (AB)_{22} & \cdots & (AB)_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
(AB)_{n1} & (AB)_{n2} & \cdots & (AB)_{np} \\
\end{pmatrix}
;]

Where [; (AB)_{ij} = \sum_{k=1}^m A_{ik}B_{kj} ;].

But how is this derived?

6) The above does not look at all like repeated addition. What does the output of matrix multiplication represent?

7) What is a determinant in human-speak?

8) What are some concrete examples of matrix algebra in the real world?

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