Rank and Trace of a Matrix

Question

For a real matrix $A$, $A$ satisfies $A^3=A$, $A$ not equal to $I$ or $O$. If $\text{Rank}(A) =r$ and $\text{Trace}(A)=t$ then,

A) $r \geq t$ and $r+t$ is odd

B) $r \geq t$ and $r+t$ is even

C) $r<t$ and $r+t$ is odd

D) $r<t$ and $r+t$ is even

I know that if A^2=k.A , then Trace (A) =k . Rank (A) , but i just don't know how to apply this here . Also tried to do A^3=A or ,A^3-A=O or , A(A^2-I)=O therefore A^2-I=O , i.e A^2=I ,therefore Trace of A^2 is n (assuming A is nxn) then ? How to calculate rank of A and connect it to trace ?

Please help.

Thank you :)

Answer 1

Try to compute the minimal polynomial of $A$ from the equation $A^3=A$ and the fact that $A$ is neither $I$ nor $O$. Then what are the two non zero eigenvalues?

Now write down $t$, $r$ and $t+r$ in terms of the multiplicity of these two eigenvalues and the answer should be clear.