Question

Find all of the eigenvalues λ of the matrix A. (Hint: Use the method of Example 4.5 of finding the solutions to the equation 0 = det(A − λI). Enter your answers as a comma-separated list.) A = 3 5 8 0

Answer 1

`\lambda=8,\ \lambda=-5`

Explanation:

Eigenvalues of a Matrix

Given a matrix A, the eigenvalues of A, called
`\lambda` are scalars who comply with the relation:

`det(A-\lambda I)=0`

Where I is the identity matrix

`I=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]`

The matrix is given as

`A=\left[\begin{array}{cc}3&5\\8&0\end{array}\right]`

Set up the equation to solve

`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`

Expanding the determinant

`det\left(\left[\begin{array}{cc}3-\lambda&5\\8&-\lambda\end{array}\right]\right)=0`

`(3-\lambda)(-\lambda)-40=0`

Operating Rearranging

`\lambda^2-3\lambda-40=0`

Factoring

`(\lambda-8)(\lambda+5)=0`

Solving, we have the eigenvalues

`\boxed{\lambda=8,\ \lambda=-5}`