Matrices!
Consider all pairs of non-singular matrices ( A , B ) such that A B = B A 4 and B 6 = I .
What is the minimum positive integer value of p , such that A p is always the identity matrix?
Bonus: Generalize your answer.
The answer is 4095.
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1 solution
Why can't we have p = 8190?
What you have shown here is that p = m n − 1 would work, but not that this is the only number that would work.
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Sorry, edited.
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This question is interesting, but I have some concerns about the phrasing / validity.
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How do we know that is the minimum value? Why does no smaller value work?
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Also, I believe you want the minimum positive value (otherwise 0 would work). In this case, notice that the general formula doesn't apply for m = 1 or n = 0 .
I was looking at these cases when trying to figure out why we have a minimum, and it's not immediately obvious that's true, esp for specific matrices. It might be true "over all pairs of matrices that satisfy the condition". (IE the matrices A = B = I satisfy the condition that A B = B A 4 , B 6 = I , but the minimum value of A n = I would be n = 1 ). -
It seems like you want them to be square matrices.
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@Calvin Lin – 1) Actually I think there are a number of matrices that satisfy the condition, hence it is the minimum value that would work for all these values . I think it would be better if I edit it to ' Minimum positive p that would work for all matrices A satisfying the criteria ' ,right??
2) yes positive value. Sorry for that
3)Yes. I have mentioned 'Non singular matrices '
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@Sumanth R Hegde – Great. I've edited the problem to reflect this.
Note: The solution is still incomplete, in that it shows that 4095 works, but hasn't shown that no smaller value would work.
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@Calvin Lin – I agree . The only other way I see would be that 4 0 9 5 = x y , ( x , y £ N , x , y > 1 ) and A x = I . If this is true then p is not the minimum value. Instead of going about seeing if all factors of 4 0 9 5 work or not, I do not know what else to do.P perhaps you could suggest a better way Sir??
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@Sumanth R Hegde – I checked most factors.. I think 4095 is the smallest number
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@Anirudh Chandramouli – Thanks for taking the trouble. I have added a note to throw light upom the same
@Sumanth R Hegde – Right, I don't think that there is an easy way to prove this, other than finding matrices as counter examples.
@Sumanth R Hegde image is not available. Plzz upload it again
The most important property of matrices that will be used in this solution again and again is associative property of multiplication of matrices
p = m n − 1 .
For m = 4 , n = 6 ...We get p = 4 0 9 5
( Note :I have shown p = m n − 1 is a solution . However, to show that it is the minimum value, one would have to check the factors of m n − 1 to rule out possibility of any other value satisfying the given criteria)