Suppose a real matrix A satisfies (A^3)=A,A is not equal to I[identity matrix] or 0[null matrix].
If rank((A))=r and trace((A))=t ,then –
a)r>=t and r+t is odd
b)r>=t and r+t is even
c)r<t and r+t is odd
d)r<t and r+t is even
Let us focus on the given information: A^3=A and A is not equal to I(The Identity matrix) and also A is not equal to O(The Null matrix)
i.e., either A=O or A^2-I=O but it is already given to us that A is not equal to O , Hence we have A^2-I=O, i.e., A^2=I which implies that A=+I,-I.
Again we have that A is not equal to +I, therefore we take A=-I which implies
rank(A)=r=n(If A and I is of order n) and trace(A)=t=-n (adding -1, n times). Hence we arrive at r+t=0 which is even and r>=t essentially.
Therefore option b) is the correct one.