Consider the matrix

• $P^2 = P$, so it's a projection. It projects the vector [x;y] to the vector [x+y;0]. Clearly, applying it twice will result in 0. [x; y], 1. [x+y; 0], 2. [x+y; 0].
• It projects the value (x+y)onto the x axis,and kills the y axis.
• It's not a projection onto the coordinate axis.