Mary asked her mother to find non-constant polynomial P(x), Q(x),R(x) and S(x) with integer coefficients such that `[[P(x),Q(x)],[R(x),S(x)]] ^2 ` = `[[1,0],[0,1]]`. What are some possible polynomials for these conditions?

A possible collection of such polynomials is `P ( x ) = x , ` `S ( x ) = -x , ` `Q ( x ) = 1 - x , ` `R ( x ) = 1 + x .`

Expert Answers

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Let's square the matrix `[ [ P , Q ] , [ R , S ] ] ` to obtain some equations:

`[ [ P , Q ] , [ R , S ] ]^2 = [ [ P , Q ] , [ R , S ] ] [ [ P , Q ] , [ R , S ] ] = [ [ P^2 + QR , P Q + Q S ] , [ P R + R S , Q R + S^2 ] ] = [ [ 1 , 0 ] , [ 0 , 1 ] ] .`

This means that `P Q + Q S = 0 , ` or `P + S = 0 ` because `Q ` is non-constant. Thus, `S = -P .`

Also, `P^2 + QR = 1 . ` Start from `P = x , ` then `Q R = 1 - x^2 = ( 1 - x ) ( 1 + x ) . ` Then we can set `Q = 1 - x ` and `R = 1 + x . ` Finally, `S = -P = -x .`

This way we obtained an example of such polynomials, `P ( x ) = x , ` `S ( x ) = -x ,` `Q ( x ) = 1 - x , ` `R ( x ) = 1 + x .`

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