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To find the eigenvalues and eigenvectors of a matrix, we typically follow these steps: 1. Find the eigenvalues by solving the characteristic equation \(\det(A - \lambda I) = 0\), where \(A\) is the matrix, \(\lambda\) is the eigenvalue, and \(I\) is the identity matrix. 2. Find the eigenvectors by solving the equation \((A - \lambda I)\mathbf{v} = 0\) for each eigenvalue \(\lambda\), where \(\mathbf{v}\) is the eigenvector. Let's apply these steps to the given matrix: \[ A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix} \] ### Step 1: Find the Eigenvalues The characteristic equation is given by: \[ \det(A - \lambda I) = 0 \] Substitute \(A\) and \(I\): \[ _______ ____ ________ _______ _____.
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