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Can someone explain the following to me:"... the entries of the matrix [tex]A - tI_n[/tex] are not scalars in the field F. They are, however, scalars in another field F(t), the field of quotients of polynomials in t with coefficients from F."

I asked someone earlier today, and I got an explanation that went something like:Entries of a matrix are in the field but since t is a variable, the diagonals of [tex]A - tI_n[/tex] are polynomials F(t). And we know the ring of polynomials doesn't necessarily have an inverse. So it follows that [tex]A - tI_n[/tex] are not scalars in the field

Questions:Can someone explain to me how the ring of polynomials not having inverses implies that [tex]A - tI_n[/tex] are not scalars in the field? Or even how the diagonals being polynomials implies such a conclusion.

Thanks,

JL

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