Matrix with same image and kernel Does exists a matrix A for which kernel of A is the same as the image of A?
Answer is True. But I couldn't find the example. I think I saw it from somewhere but I can't find it.  It was 2 by 2 matrix. 

Let $A = e_1 e_2^T$. Then $\ker A = \operatorname{sp} \{ e_1 \}$, and ${\cal R }(A) = \operatorname{sp} \{ e_1 \}$.
Note that since since $\dim {\cal R }(A) + \dim \ker A = n$, the dimension of the domain must be even.
