Does there exist a non zero $2 \times 2$ matrix $A$ such that $A^2 \neq 0$ and $A^3=0$, where $0$ is the $2 \times 2$ zero matrix?
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Chinnapparaj R
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Vipra Dosi
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Welcome to MSE! For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – platty Nov 30 '18 at 05:24
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@Vipra: Interrelate nilpotence and cayley Hamilton! – Chinnapparaj R Nov 30 '18 at 05:29
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If such a matrix exist,$f(x)=x^3$ is an annihilating polynomial for $A$. Since minimal polynomial divides annihilating polynomials, $m(x)=x$ or $x^2$. Note that $m(x)\neq x^3$, since the characteristic polynomial is of degree $2$. If $m(x)=x$,we have $A =0$ and if $m(x)=x^2$,we have $A^2 =0$.
Hence such a matrix doesn't exist.
cqfd
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