Need help with a Matrix problem

Question

I need help finding an example of a matrix that is a nonzero, non-identity matrix A such that A^2 = A. and show that it is non-invertible.

Could someone please explain what this means?

Thanks.

Answer 1

An example is $A=\pmatrix{1&0\\0&0}$.

If $A^2=A$ with invertible $A$ you get $A=A^2 A^{-1}=AA^{-1}=E$. Thus the identity matrix is the only invertible $A$ such that $A^2=A$.