Question

4. Find the sum and product of the eigenvalues of the matrix

2 3 -2
-2 1 1
1 0 2

Answer 1

Given matrix is:$\begin{bmatrix} 2 & 3 & -2 \\ -2 & 1 & 1\\ 1 & 0 & 2 \end{bmatrix}$

Now, solving $\begin{vmatrix} 2-\lambda & 3 & -2 \\ -2 & 1-\lambda & 1\\ 1 & 0 & 2-\lambda \end{vmatrix}=0$ , we get:

$\Rightarrow(2-\lambda)[(1-\lambda)(2-\lambda)-0]-3[-2(2-\lambda)-1]-2[0-1(1-\lambda)]=0\\ \Rightarrow (1-\lambda)(2-\lambda)^2+12-6\lambda+3+2-2\lambda=0\\ \Rightarrow (1-\lambda)(2-\lambda)^2+17-8\lambda=0\\ \Rightarrow 4+\lambda^2-4\lambda-4\lambda-\lambda^3+4\lambda+17-8\lambda=0\\ \Rightarrow -\lambda^3+\lambda^2-12\lambda+21=0\\ \Rightarrow \lambda^3-\lambda^2+12\lambda-21=0\\$

So, sum of eigenvalues of matrix is: $-(\frac{-1}{1})=1$

and product of eigenvalues of matrix is: $-(\frac{21}{1})=-21$