Let $A$ be a square matrix such that $A^2 = A$. Show that $A$ cannot be a strictly diagonally dominated matrix unless A is the identity matrix.

Question

Let $A$ be a square matrix such that $A^2 = A$. Any idea how to show that $A$ cannot be a strictly diagonally dominated matrix unless $A$ is the identity matrix.

Answer 1

Write the equality as $A(A-I)=0$ and use the fact that strictly diagonally dominated matrices are invertible (you can prove that, right?) to eliminate $A$.