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Let $A$ be an $n\times n, (n\ge2)$ matrix with char poly $x^{n-2}(x^2-1)$ Then which of The following is true?

  1. $A^n=A^{n-2}$,

  2. $r(A)=2$,

  3. $r(A)$ is atleast $2$,

  4. there exist non zero vector $x,y$ such that $A(x+y)=x-y$,

Well, I can only see that 1 is true from the caley Hamilton Theorem, would you help other three are correct or not?

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    @alex.jordan Of course, my bad. That is what I meant.2012-06-05

1 Answers 1

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Since $-1$ and $1$ are eigenvalues of $A$, the rank of $A$ is at least $2$. But it can be more than $2$, for example if $J$ is the matrix of size $(n-2)\times (n-2)$ given by $J=\pmatrix{0&1&0&\ldots &0\\ 0&0&1&\ldots &0\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ 0&0&\ldots&0&1\\ 0&0&\ldots&0&0}$ and $A:=\pmatrix{A'&0\\ 0&J}$, where $A=\pmatrix{1&0\\-1&0}$. (in fact the rank is $2$ if $n=2$, and between $2$ and $n-1$ when $n>2$.

For the last question, use eigenvectors for the eigenvalues $-1$ and $1$.

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    For the last question, you could also just use any element of the kernel, and set $x = y$. (edit: oops, as long as n > 2, I mean)2012-06-04