On determinant, other characteristic polynomial coefficients, and inverses in Clifford algebras of arbitrary dimension
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In this paper, we present the formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverses in real Clifford algebras over vector space of arbitrary dimension n using only the operations of multiplication, summation, and operations of conjugation without using the corresponding matrix representations. We use the methods of Clifford algebras (including the method of quaternion typification suggested by the author in previous papers and the method of operations of conjugation of special type presented in this paper) and the matrix methods (related to the Cayley-Hamilton theorem, the Faddeev-LeVerrier algorithm, and the Bell polynomials), which we generalize to the case of Clifford algebras in this paper. We present the construction of operations of conjugation of special type and study relations between these operations and the projection operations onto fixed subspaces of Clifford algebras. We use this construction in the analytical proof of the formulas for the determinant, other characteristic polynomial coefficients, adjugate, and inverses in Clifford algebras.
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