Given the matrix \( A = \begin{bmatrix} 1 & 3 \ 4 & 2 \end{bmatrix} \), find the characteristic equation and hence the eigenvalues.
Answer
Setting this equation equal to zero gives the characteristic equation, which is \( \lambda^2 - 3\lambda - 10 = 0 \).
Step 1 ：The characteristic equation is obtained by setting the determinant of \( (A - \lambda I) \) equal to zero, where \( I \) is the identity matrix and \( \lambda \) are the eigenvalues. Hence, \( A - \lambda I = \begin{bmatrix} 1-\lambda & 3 \ 4 & 2-\lambda \end{bmatrix} \).
Step 2 ：We then find the determinant of \( A - \lambda I \), which is \( (1-\lambda)(2-\lambda) - (3)(4) = \lambda^2 - 3\lambda - 10 \).
Step 3 ：Setting this equation equal to zero gives the characteristic equation, which is \( \lambda^2 - 3\lambda - 10 = 0 \).
