ISI2016-MMA-28

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

Comments

How $A^{3}=0$ implies $A^2=0$ ?

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It doesn't implies but it is given that A^3 is 0 then it implies either A^2 or A is zero.

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Does it really ?

 $A=\begin{pmatrix} 0 &0 &0 \\ 1&0 &0 \\ 0& 1 &0 \end{pmatrix}$

try for this .