Cayley-Hamilton Theorem/Matrix

Theorem
Let $A$ be a commutative ring with unity.

Let $\mathbf N = \left({a_{ij} }\right)$ be an $n \times n$ matrix with entries in $A$.

Let $\mathbf I_n$ denote the $n \times n$ unit matrix.

Let $p_N \left({x}\right)$ be the determinant $\det \left({x \cdot \mathbf I_n - \mathbf N}\right)$.

Then:
 * $p_N \left({N}\right) = \mathbf 0$

as an $n \times n$ zero matrix.

That is:


 * $ N^n + b_{n-1} N^{n-1} + \cdots + b_1 N + b_0 = \mathbf 0$

where the $b_i$ are the coefficients of $p_N \left({x}\right)$.

Proof
Taking $\phi = N$ in the proof of Cayley-Hamilton Theorem for Finitely Generated Modules we see that $N$ satisfies:


 * $p_N \left({x}\right) = \det \left({x \cdot I_n - N}\right) = 0$

Take $\mathfrak a$ to be the ideal generated by the entries of $N$.