Answer
The eigenvalues of the given matrix A are $\lambda_1=-3+2\sqrt 2; \lambda_2=-3-2\sqrt 2$
Work Step by Step
We are given:
$\det (A- \lambda I)=0$
Using cofactor expansion along the first column of $A−\lambda I$, we obtain
$\det(A−\lambda)=\det \begin{bmatrix}
-1-\lambda & 2\\
-4 & 7-\lambda
\end{bmatrix}$
$=(-1- \lambda).(7- \lambda)-2.(-4)$
$=-7+\lambda -7\lambda +\lambda^2 +8$
$=\lambda^2 -6\lambda +1$
$\rightarrow \lambda_1=-3+2\sqrt 2; \lambda_2=-3-2\sqrt 2$
The eigenvalues of the given matrix A are $\lambda_1=-3+2\sqrt 2; \lambda_2=-3-2\sqrt 2$
