Chapter E, Exercise EE.C22
Without using a calculator, find the eigenvalues of the matrix B.
\[ B = \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix} \]
Solution
To find the eigenvalues of A, we must solve det(λI-B) =0 for λ.Where I is Identitity Matrix. \[ det(λ\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}) = 0 \] \[ det(\begin{bmatrix} λ & 0 \\ 0 & λ \end{bmatrix} - \begin{bmatrix} 2 & -1 \\ 1 & 1 \end{bmatrix}) = 0 \]
\[ det(\begin{bmatrix} λ-2 & 1 \\ -1 & λ-1 \end{bmatrix}) = 0 \] \[(λ-2)(λ-1) + 1 = 0\]
\[ λ^2 -λ -2λ +2 + 1= 0 \]
\[ λ^2 -3λ +3 = 0 \] Here, We need to solve the quadratic formula The general formula is:-
\[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
Applying the same in our equation \[λ=\frac{-(-3)\pm\sqrt{(-3)^2-4*3}}{2*1}\]
\[λ=\frac{3\pm\sqrt{9-12}}{2}\] \[λ=\frac{3\pm\sqrt{-3}}{2}\] or
\[λ=\frac{3\pm\sqrt{3i}}{2}\] Hence the 2 values for λ or eigenvaulues are
\[λ=\frac{3+\sqrt{3i}}{2}\] and
\[λ=\frac{3-\sqrt{3i}}{2}\]