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Let A be a square matrix such that $A^{3}$ = 0, but $A^{2} \neq 0$. Then which of the following statements is not necessarily true?

(A) $A \neq A^{2}$

(B) Eigenvalues of $A^{2}$ are all zero

(C) rank(A) > rank($A^{2}$ )

(D) rank(A) > trace(A)
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Option C can never be true
rank (A*B) = min {r(A),r(B)}
so rank (A*A) = r(A)


Option A is true 
suppose A2 = A   now
A3 = A2*A
     = A*A
     =A2 

and given A3 is 0 this implies A2 is also zero this contradicts statement given in question A2 not eqal to zero 
so our assumption A2 = A is wrong . this implies A2 is not equal to A
For other options i dont have exact answer :)

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