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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

Consider the polynomial

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 Student Mathematica Contest.

Determinant (Wolfram MathWorld)

Matrix Trace (Wolfram MathWorld)

Our First Russian Student Competition (Wolfram Blog)

"The Determinant Using Traces" from the Wolfram Demonstrations Project
http://demonstrations.wolfram.com/TheDeterminantUsingTraces/

Contributed by: Oleksandr Pavlyk

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