ISI Sample Paper Question

Question

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)

Answer 1

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 :)

Comments

Isn't there a correction because
rank (A*B) <= min {r(A),r(B)}

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option B will not necessarily true as mentioned that A^2 NOT Zer0 therefore all of its eigen value must be non zero otherwise product of eigen values results zero that leads to  A^2 =0 which is not desired