We prove the bivariate Cayley-Hamilton theorem, a powerful generalization of the classical Cayley-Hamilton theorem. The bivariate Cayley-Hamilton theorem has three direct corollaries that are usually proved independently: The classical Cayley-Hamilton theorem, the Girard-Newton identities, and the fact that the determinant and every coefficient of the characteristic polynomial has polynomially sized algebraic branching programs (ABPs) over arbitrary commutative rings. This last fact could so far only be obtained from separate constructions, and now we get it as a direct consequence of this much more general statement. The statement of the bivariate Cayley-Hamilton theorem involves the gradient of the coefficient of the characteristic polynomial, which is a generalization of the adjugate matrix. Analyzing this gradient, we obtain another new ABP for the determinant and every coefficient of the characteristic polynomial. This ABP has one third the size and half the width compared to the current record-holder ABP constructed by Mahajan-Vinay in 1997. This is the first improvement on this problem for 28 years. Our ABP is built around algebraic identities involving the first order partial derivatives of the coefficients of the characteristic polynomial, and does not use the ad-hoc combinatorial concept of clow sequences. This answers the 26-year-old open question by Mahajan-Vinay from 1999 about the necessity of clow sequences. We prove all results in a combinatorial way that on a first sight looks similar to Mahajan-Vinay, but it is closer to Straubing's and Zeilberger's constructions.

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