Problem

Given the matrix \( A = \begin{bmatrix} 1 & 3 \ 4 & 2 \end{bmatrix} \), find the characteristic equation and hence the eigenvalues.

Solution

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 \( \lambda^2 - 3\lambda - 10 = 0 \).

From Solvely APP
Source: https://solvelyapp.com/problems/3OakYsa4N0/

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