2025 USA-NA-AIO Round 1, Problem 1, Part 5

AI Olympiads 2025 USA-NA-AIO Round 1
1

Part 5 (10 points, non-coding task)

Let us go back to our matrix \mathbf{A}.

In the following eigenvalue equation

\mathbf{A} \mathbf{x} = \lambda \mathbf{x} ,

compute two eigenvalues \lambda_0 and \lambda_1 whose values are in a descending order.

  • Reasoning is required.
2

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We solve the following characteristic equation:

\det \left( \mathbf{A} - \lambda \mathbf{I} \right) = 0 .

Thus, we compute \lambda that satisfies

\left( 1 - \lambda \right) \left( - \lambda \right) - 1 \cdot 1 = 0 .

Hence, all eigenvalues are

\boxed{\lambda_0 = \frac{1 + \sqrt{5}}{2} , \quad \lambda_1 = \frac{1 - \sqrt{5}}{2} .}

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