  Pairs of $n\times n$ matrices whose commutator differ from the identity by a
matrix of rank $r$ are used to construct bispectral differential operators with
$r\times r$ matrix coefficients satisfying the Lax equations of the Matrix KP
hierarchy. Moreover, the bispectral involution on these operators has dynamical
significance for the spin Calogero particles system whose phase space such
pairs represent. In the case $r=1$, this reproduces well-known results of
Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to
the bispectrality of (scalar) differential operators. This new class of pairs
$(L, \Lambda)$ of bispectral matrix differential operators is different than
those previously studied in that $L$ acts from the left, but $\Lambda$ from the
right on a common $r\times r$ eigenmatrix.
