Answer:
[tex]\lambda=8,\ \lambda=-5[/tex]
Step-by-step explanation:
Eigenvalues of a Matrix
Given a matrix A, the eigenvalues of A, called [tex]\lambda[/tex] are scalars who comply with the relation:
[tex]det(A-\lambda I)=0[/tex]
Where I is the identity matrix
[tex]I=\left[\begin{array}{cc}1&0\\0&1\end{array}\right][/tex]
The matrix is given as
[tex]A=\left[\begin{array}{cc}3&5\\8&0\end{array}\right][/tex]
Set up the equation to solve
[tex]det\left(\left[\begin{array}{cc}3&5\\8&0\end{array}\right]-\left[\begin{array}{cc}\lambda&0\\0&\lambda \end{array}\right]\right)=0[/tex]
Expanding the determinant
[tex]det\left(\left[\begin{array}{cc}3-\lambda&5\\8&-\lambda\end{array}\right]\right)=0[/tex]
[tex](3-\lambda)(-\lambda)-40=0[/tex]
Operating Rearranging
[tex]\lambda^2-3\lambda-40=0[/tex]
Factoring
[tex](\lambda-8)(\lambda+5)=0[/tex]
Solving, we have the eigenvalues
[tex]\boxed{\lambda=8,\ \lambda=-5}[/tex]
