  We consider Fredholm determinants of matrix convolution operators associated
to matrix versions of the $n - $th Airy functions. Using the theory of
integrable operators, we relate them to a fully noncommutative Painlev\'e II
hierarchy, defined through a matrix valued version of the Lenard operators. In
particular, the Riemann-Hilbert technique used to study these integrable
operators allows to find a Lax pair for each member of the hierarchy. Finally,
the coefficients of the Lax matrices are explicitely written in terms of these
matrix valued Lenard operators and some solution of the hierarchy are written
in terms of Fredholm determinants of the square of the matrix Airy convolution
operators.
