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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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    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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    @AlkaGoyal no.It is right..2013-06-06