Doubt over the proof of Cayley- Hamilton heorem 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 $\lambda$)
$Q(\lambda) =Q_0 + Q_1\lambda + Q_2 \lambda^2  +\ldots Q_n \lambda^n$. 
Thank you

If $Q(\lambda)$ is the adjoint of $P-\lambda I$, by definition $Q(\lambda)(P-\lambda I)=det(P-\lambda I) I$.
Compare the maximum degree element of each matrix of the equality. If an element $q_{i,j}$ of $Q(\lambda)$ had degree $k$, then element in each diagonal element of the matrix would have max degree element would have degree $k+1$. You can prove it writing down the product on the left. So $n=k+1$.
