Cayley-Hamilton Theorem/Matrices

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

Let $\mathbf N = \sqbrk {a_{i j} }$ be an $n \times n$ matrix with entries in $A$.

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

Let $\map {p_{\mathbf N} } x$ be the determinant $\map \det {x \cdot \mathbf I_n - \mathbf N}$.

Then:
 * $\map {p_{\mathbf N} } {\mathbf N} = \mathbf 0$

as an $n \times n$ zero matrix.

That is:


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

where the $b_i$ are the coefficients of $\map {p_{\mathbf N} } x$.

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


 * $\map {p_{\mathbf N} } x = \map \det {x \cdot \mathbf I_n - \mathbf N} = 0$

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