I'm always getting confused about how matrix multiplication works. Here, you can try it interactively:

Note that the matrix dimensions are entered as rows x columns (height x width), and the element indices are also written like this. Click source matrix elements to toggle between zero (white) and nonzero (orange).

Note that A^TA = AA^T, for square and symmetric A. It is not true for other matrices (even though result of A^TA or AA^T is symmetric - which is always btw).

Note that a single column of AA^T scatters to the whole product matrix, but the columns are independent (there are no product between elements from different columns of A in AA^T).

Link to the older version of this applet.