By Flanigan & Kazdan: some matrices http://38.media.tumblr.com/tumblr_lssm6aGqk81qc38e9o1_1280.png some more matrices http://33.media.tumblr.com/tumblr_lssm6aGqk81qc38e9o2_1280.png
Instead of looking at a "box of numbers", look at the "total action" after applying the whole thing. It's an automorphism of linear spaces, meaning that in some vector-linear-algebra-type situation this is "turning things over and over in your hands without breaking the algebra that makes it be what it is". (Modulo some things—like maybe you want a constant determinant.)
determinant--drawn by me http://media.tumblr.com/tumblr_ljzkimel1e1qbdydv.png
This is also why order matters: if you compose the matrices in one direction it might not be the same as the other. $$^1_4 \Box ^2_3 {} \xrightarrow{\mathbf{V} \updownarrow} {} ^4_1 \Box ^3_2 {} \xrightarrow{\Theta_{90} \curvearrowright} {} ^1_2 \Box ^4_3 $$ versus $$^1_4 \Box ^2_3 {} \xrightarrow{\Theta_{90} \curvearrowright} {} ^4_3 \Box ^1_2 {} \xrightarrow{\mathbf{V} \updownarrow} {} ^3_4 \Box ^2_1 $$
The actions can be composed (one after the other)—that's what multiplying matrices does. Eventually the matrix representing the overall cumulative effect of whatever things you composed, should be applied to something. For this you can say "the plane", or pick a few points, or draw an F, or use a real picture (computers are good at linear transformations after all).
mona lisa shear http://upload.wikimedia.org/wikipedia/commons/3/3c/Mona_Lisa_eigenvector_grid.png the 3-D version http://www.kidsmathgamesonline.com/images/pictures/shapes/parallelepiped.jpg
You could also watch the matrices work on Mona step by step too, to help your intuition.