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How can I find the inverse of $A$ using Cayley Hamilton Theorem?

    A=    0 1 0 0
          0 0 1 0
          0 0 0 1
          1 0 0 0

The Characteristic equation of $A$, I get is $A^4=0$, which implies $A=0$ which is clearly not true. Please help.

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    Two comments: first of all $A$ seems not a square matrix, how do you evaluate the determinant (and the characteristic polynomial)? Also, if $A$ is a matrix, $A^4=0$ does not necessarily imply $A=0$. – 2017-02-09
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    You have the wrong characteristic equation. Check your algebra. – 2017-02-09
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    Also, if it was the case that $A^4=0$ (it's not, but suppose it was), then what could you say about $\text{det}(A)$? Based on your answer to that, what could you say about the invertibility of $A$? – 2017-02-09
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    @quasi thank you. It was a stupid mistake. So I got A^3=A^(-1). So for inverse I would have to find cube of matrix A right? There is no shortcut around that is there? – 2017-02-09
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    It's not the worst thing in the world. – 2017-02-09
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    Follow the previous question of @quasi. Does the inverse always exist? – 2017-02-09
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    @quasi No not by far. Thanks! – 2017-02-09
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    A characteristic equation A^n=0 will always imply A=0, because multiplication by a non singular matrix does not change rank of the matrix. – 2017-02-09
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    Computing $A^3$ is easy. Observe that left-multiplying a vector by $A$ rotates that vector’s components one slot “up,” so performing this three times does ...? – 2017-02-09
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    @amd 3 slots up. Yay, thanks! – 2017-02-10

1 Answers 1

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I'll expand my above comment. Suppose we have a $4$-dimensional square matrix $A$ (which is not the zero matrix) with characteristic polynomial $p(\lambda)=\lambda^4$; as I previously said if $B$ is a matrix, $B^n=0$ does not imply in general that $B$ is the zero matrix. Indeed, Cayley-Hamilton itself provides examples of this fact: take our matrix $A$, we have $p(A)=A^4=0$, while $A$ is non-zero (perform a direct calculation if you're sceptic).

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    Answer based on the previous (wrong) version of the post. – 2017-02-09