Find the largest eigenvalue of A = [4 1; 2 3].

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Find the largest eigenvalue of $A = [4213]$ .

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Hint

For a $2 \times 2$ matrix the eigenvalues are fixed by the trace and the determinant.

Solution

The trace is $4 + 3 = 7$ and the determinant is $4 \times 3 - 1 \times 2 = 10$ .
The eigenvalues solve $det (A - λ I) = λ^{2} - 7 λ + 10 = 0$ , which factors as $(λ - 2) (λ - 5) = 0$ .
As a check, $A (1, 1) = (5, 5)$ , so $(1, 1)$ is an eigenvector for $5$ .
The largest eigenvalue is $5$ .

Answer: $5$