How does a 2x2 matrix transform a point?

Drag the blue point about to see what transformation the pink matrix A will have on it. See if you can find the lines through the centre where, if you are on the line, you will stay somewhere on that line. Click the boxes to check if you've found them. Change the matrix A in the spreadsheet to see what happens.
Will you always get two lines through the origin? Why are the lines through the origin? Can you find a matrix where the point stays exactly where is is?