# Trying to find principal minor of a matrix to get charecteristic polynomial

I have a quick question I am trying to get the charecterstic polynimal of the following matrix using the principal minors of the matrix.

Using the following polynomial

$P_A(t)=t^4-E_1(A)t^3+E_2(A)t^2-E_3(A)t+E_4(A)$

I know the characteristic polynomial is

$P_A(t)=t^4-4t^3+3t^2+2t-1$

I know $E_1(A)=4$ is the trace

and $E_4(A)=-1$ the determinant

but I am having trouble choosing 4 principal minor in E_3(A) so they add up to -2.

$$\begin{pmatrix} 1 & 1 & 0 &0 \\ 1 & 1& 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 &1 \end{pmatrix}$$

$E_k(A)$ is the sum of $k\times k$ principal minors of $A$, e.g., $E_3(A)=\det\left[\begin{matrix}1&1&0\\1&1&1\\0&1&1\end{matrix}\right]+0+0+\det\left[\begin{matrix}1&1&0\\1&1&1\\0&1&1\end{matrix}\right]=-1-1=-2$.