# Find nonconstant polynomials 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]]`.

A matrix `[[P(x), 1-P(x)], [1+P(x), -P(x)]] ` will work for any nonconstant polynomial P with integer coefficients. Matrix multiplication is performed by multiplying rows of the left matrix by the columns of the second matrix. Here both the left and the right matrix is `[ [ P , Q ] , [ R , S ] ] .`

Consider the element at the first row and the...

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Matrix multiplication is performed by multiplying rows of the left matrix by the columns of the second matrix. Here both the left and the right matrix is `[ [ P , Q ] , [ R , S ] ] .`

Consider the element at the first row and the first column of the product:`[ [ P , Q ] ] [ [ P ] , [ R ] ] = P^2 + QR . ` And this polynomial must be equal to `1 ` for any value of `x .`

Rewrite this as `QR = 1 - P^2 = ( 1 - P ) ( 1 + P ) . ` Let's try to set `Q = 1 - P , ` `R = 1 + P , ` then the equation is satisfied, and if `P ` is nonconstant, then `Q ` and `R ` are nonconstant, too. Also, if `P ` has integer coefficients, so do `Q ` and `R.`

Now consider the element at the first row and the second column of the product: `[ [ P , Q ] ] [ [ Q ] , [ S ] ] = PQ + QS = Q ( P + S ) ` and it must be zero. For this, it is sufficient that `P + S = 0 , ` or `S = -P . ` Again, `S ` will be nonconstant for a nonconstant `P ` and will have integer coefficients if `P ` has integer coefficients.

The other two cells will give the equations `R(P+S) = 0, ` which is already satisfied because `S = -P, ` and `RQ + S^2 = 1,` which is already satisfied for the same cause.

This way, we may choose any nonconstant polynomial with integer coefficients `P ` and let `Q = 1 - P, ` `R = 1 + P, ` `S = -P ` to satisfy the matrix equation.

For example, `P(x) = x - 2, ` `Q(x) = 3 - x, ` `R(x) = x - 1, ` `S(x) = 2 - x.`

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