Find the characteristic polynomial of the matrix \(A=\begin{bmatrix} 3 & 2 & 1\\ 0 & 1 & 1\\ 1 & 2 & 0\end{bmatrix}\)
To find the characteristic polynomial, we need to find \(\lambda I_3 - A\) where \(\lambda\) is the eigenvalue, \(I_3\) is the \(3\times3\) identity matrix and A is the given matrix
\(\Rightarrow \lambda I_3 - A = \left(\begin{bmatrix} \lambda & 0 & 0\\ 0 & \lambda & 0\\ 0 & 0 & \lambda\end{bmatrix}-\begin{bmatrix} 3 & 2 & 1\\ 0 & 1 & 1\\ 1 & 2 & 0\end{bmatrix}\right)\)
\(\Rightarrow det(\lambda I_3 - A)= det\begin{bmatrix} \lambda-3 & -2 & -1\\ 0 & \lambda-1 & -1\\ -1 & -2 & \lambda\end{bmatrix}\)
\(det(\lambda I_3 - A)= (\lambda-3)(\lambda-1)(\lambda) - 2 - 2(\lambda-3) - (\lambda-1)\)
\(det(\lambda I_3 - A)= (\lambda^2 - 4 \lambda +3) \lambda -2 -2 \lambda + 6 - \lambda +1\)
\(det(\lambda I_3 - A)= \lambda^3 -4 \lambda^2 +3 \lambda + 5 -3 \lambda\)
\(p(x) = \lambda^3 - 4 \lambda^2 + 5\)
\(p(x)\) is the characteristic polynomial of the matrix \(A=\begin{bmatrix} 3 & 2 & 1\\ 0 & 1 & 1\\ 1 & 2 & 0\end{bmatrix}\)
Suppose that A is a square matrix. Prove that the constant term of the characteristic polynomial of A is equal to the determinant of A.
Let A be a square matrix with 2 dimension \(2\times2\)
\(A=\begin{bmatrix} a & b\\ c & d\end{bmatrix}\)
To prove that the constant of the characteristic polynomial is equal to the determinant of A, let’s find the characteristic polynomial of A first:
\(\Rightarrow det(\lambda I_2 - A)= \left(\begin{bmatrix} \lambda & 0\\ 0 & \lambda\end{bmatrix} - \begin{bmatrix} a & b\\ c & d\end{bmatrix}\right)\)
\(\Rightarrow det(\lambda I_2 - A)= \left(\begin{bmatrix} \lambda-a & -b \\ -c & \lambda-d\end{bmatrix}\right)\)
\(\Rightarrow det(\lambda I_2 - A)= (\lambda-a) (\lambda-d) - (cb)\)
\(\Rightarrow det(\lambda I_2 - A)= \lambda^2 -d \lambda - a \lambda + ad - cb\)
\(\Rightarrow p(x) = \lambda^2 - \lambda(d+a) + (ad-cb)\)
The constant of \(p(x)\) is equal to \((ad-cb)\) and \(det(A)=ad-cb\), So the constant of \(p(x)\) and the determinant of the matrix \(A=\begin{bmatrix} a & b\\ c & d\end{bmatrix}\) are equal.