Consider the matrix $A = \begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$. What is the largest eigenvalue $\lambda_1$ of matrix $A$?

Answer Key:

5

### Solution: Characteristic equation: $\det(A - \lambda I) = (4-\lambda)(3-\lambda) - 2 = \lambda^2 - 7\lambda + 12 - 2 = \lambda^2 - 7\lambda + 10 = 0$. Roots: $(\lambda - 5)(\lambda - 2) = 0 \implies \lambda = 5, 2$. Largest eigenvalue is 5. Correct: 5.

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