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Consider a matrix$ A = (a_{ij})_{n\times n}$ with integer entries such that $a_{ij} = 0$ for $i>j$ and $a_{ii} = 1$ for $i=1\dots n$. Which of the following properties must be true?

  1. $A^{-1}$ exists and it has integer entries
  2. $A^{-1}$ exists and it has some entries which are nt integers
  3. $A^{-1}$ is a polynomial function of $A$ with integer coefficients
  4. $A^{-1}$ is not a power of $A$ unless $A$ is an identity matrix

I am confuse about fourth option.

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    For some basic information about writing math at this site see e.g. [here](http://meta.math.stackexchange.com/questions/5020/), [here](http://meta.stackexchange.com/a/70559/155238), [here](http://meta.math.stackexchange.com/questions/1773/) and [here](http://math.stackexchange.com/editing-help#latex).2012-12-18
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    Reading your question, I wonder if instead of the condition "$a_{ij} = 1$ for $i = 1\ldots n$" you in fact want the condition "$a_{ii} = 1$ for $i = 1\ldots n$"? In which case $A$ is triangular and is identically 1 on the diagonal, so the fourth option would actually make sense as a statement.2012-12-18
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    ohh yes u r right,.2012-12-18

1 Answers 1

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Property 1. Certainly. The last property is $A^n = A^{-1}$ iff $A=I$.

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    ok,thanks sir, what is logic behid this2012-12-18
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    @Alka: why did you accept the answer if you don't understand the "logic behid this"?2012-12-18
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    option 4 is false?2012-12-18
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    @AlkaGoyal no.It is right..2013-06-06