Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over $GF(2)$ in Hrubes-Tzameret [SICOMP'15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory $\mathbf{VNC}^2$; the latter is a first-order theory corresponding to the complexity class $\mathbf{NC}^2$ consisting of problems solvable by uniform families of polynomial-size circuits and $O(\log ^2 n)$-depth. This also establishes the existence of uniform polynomial-size propositional proofs operating with $\mathbf{NC}^2$-circuits of the basic determinant identities over the integers (previous propositional proofs hold only over the two element field).

Vastly improved exposition: background on bounded arithmetic, bounded reverse mathematics, proofs have been extended, more details supplemented, more streamline flow of the construction, and minor fixes.

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over $GF(2)$ in Hrubes-Tzameret [SICOMP'15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory $\mathbf{VNC}^2$; the latter is a first-order theory corresponding to the complexity class $\mathbf{NC}^2$ consisting of problems solvable by uniform families of polynomial-size circuits and $O(\log ^2 n)$-depth. This also establishes the existence of uniform polynomial-size $\mathbf{NC}^2$-Frege proofs of the basic determinant identities over the integers (previous propositional proofs hold only over the two element field).