Editing Cayley–Hamilton theorem (section)

==== Adjugate matrices ====
All proofs below use the notion of the [[adjugate matrix]] {{math|adj(''M'')}} of an {{math|''n''&thinsp;×&thinsp;''n''}} matrix {{math|''M''}}, the [[transpose]] of its [[Minor (linear algebra)|cofactor matrix]]. This is a matrix whose coefficients are given by polynomial expressions in the coefficients of {{math|''M''}} (in fact, by certain {{math|(''n'' − 1)&thinsp;×&thinsp;(''n'' − 1)}} determinants), in such a way that the following fundamental relations hold,
<math display="block">\operatorname{adj}(M)\cdot M=\det(M)I_n=M\cdot\operatorname{adj}(M)~.</math>
These relations are a direct consequence of the basic properties of determinants: evaluation of the {{math|(''i'', ''j'')}} entry of the matrix product on the left gives the expansion by column {{math|''j''}} of the determinant of the matrix obtained from {{math|''M''}} by replacing column {{math|''i''}} by a copy of column {{math|''j''}}, which is {{math|det(''M'')}} if {{math|''i'' {{=}} ''j''}} and zero otherwise; the matrix product on the right is similar, but for expansions by rows.

Being a consequence of just algebraic expression manipulation, these relations are valid for matrices with entries in any commutative ring (commutativity must be assumed for determinants to be defined in the first place). This is important to note here, because these relations will be applied below for matrices with non-numeric entries such as polynomials.