I am having some doubt in the proof of Cayley Hamilton theorem. This theorem says that every matrix is a root if its characteristic polynomial.
Proof goes as follows:
Let us assume that matrix $A$ is of order $n\times n$. If $P(\lambda)$ be its characteristic polynomial, then by the definition of the characteristic polynomial
$P(\lambda) = det (A - \lambda I) = P_0 + P_1\lambda + P_2 \lambda^2 +\ldots P_n \lambda^n$.
Next, suppose that $Q(\lambda)$ be the adjoint matrix of $(A - \lambda I)$, such that
$Q(\lambda) =Q_0 + Q_1\lambda + Q_2 \lambda^2 +\ldots Q_k \lambda^k$.
I am not able to understand why the polynomial expression of $Q(\lambda)$ is of degree $k$? Can't I write $Q(\lambda)$ as follows (degree $n$ polynomial in $\lambda$)
$Q(\lambda) =Q_0 + Q_1\lambda + Q_2 \lambda^2 +\ldots Q_n \lambda^n$.