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The Determinant Using Traces

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The determinant of a square matrix can be computed as a polynomial of traces of the matrix and its powers. This expression greatly simplifies for traceless matrices.

Contributed by: Oleksandr Pavlyk (March 2011)
Open content licensed under CC BY-NC-SA

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Consider the polynomial in of degree , where is the identity matrix. Its leading coefficient is .

On the other hand,

.

Comparing coefficients in the powers of λ gives .

This derivation is due to Vladimir Dudchenko, the first prize winner of the Russian StudentMathematicaContest.

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