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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, here, here and here. –  Julian Kuelshammer Dec 18 '12 at 12:23
    
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. –  Willie Wong Dec 18 '12 at 13:09
    
ohh yes u r right,. –  Alka Goyal Dec 18 '12 at 13:43

1 Answer 1

up vote 1 down vote accepted

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 this –  Alka Goyal Dec 18 '12 at 12:30
2  
@Alka: why did you accept the answer if you don't understand the "logic behid this"? –  Willie Wong Dec 18 '12 at 13:05
    
option 4 is false? –  Alka Goyal Dec 18 '12 at 13:42
    
@AlkaGoyal no.It is right.. –  learner Jun 6 '13 at 17:03

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