Prove that coefficient of $λ^n$ in the characteristic polynomial is $1$.

Question

The characteristic equation is shown below

$p\left(\lambda\right) = \det\left(A-\lambda\mathbf{I}\right) = \left|\begin{array}{cccc} a_{11}-\lambda & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22}-\lambda & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn}-\lambda \end{array}\right|$

$= \left(-1\right)^n \left[\lambda^n + c_1 \lambda^{n-1} + c_2 \lambda^{n-2} + \cdots + c_{n-1} \lambda + c_n\right]$

and it is identified the coefficient of $λ^n$ to be $(-1)^n$. but I not sure how we can prove this equal to $1$ (monic)

Answer 1

Using the definition of the characteristic polynomial that you have given the coefficient will be $(-1)^n$ not $1$. Inorder to always obtain a monic polynomial you can use the definition $$p(\lambda) = det(\lambda I-A)$$ That way the roots of the polynomials are the same (i.e. the eigenvalues can still be found correctly) but other properties will differ (e.g. it isn't necessarily monic).

Answer 2

Expand the determinant in minors using Laplace's formula. You then see there is only one way of obtaining a term with $n$ powers of $\lambda$, namely multiplying the $n$ different $-\lambda$ together. You get $$(-\lambda)^n=(-1)^n \lambda^n.$$