3a, 2a1 2a 12
3a11 a3a12
'IA XA -X2 ax2
a= axx2 vX 9X2 1 1
ux8X2 aa21 aa22
xx2 a 3a22a22
a'21 a22 3x2 x2
xx2 xx2x2 1X2
xaIx2 aX1X2 X+ xi
aa2, a22 a2, o2a1 22
ax, ax, ax ax2 ax1ax2
ax2 ax2,, a,
aX2 aX, + a3 a3
a2a2, a2 a22 3a2, a322
ax- a- ax2 ax x2
dXi aXi ax2 I X2
and again, differentiation of any row with respect to a variable upon which it does not
depend introduces an all zero row. Thus, if in the preceding result the first row of A is
independent of x2 and the second of x,, then only one of the determinants in the expan-
sion survives
a2a11 a2a12
alI a4 aX2
3x23x2 3a2, 3a22
3x2 3x2
Since each P(i I n) appears in only one row of -I P; 1| = I P I -1 -II, it fol-
lows from application of the preceding determinant differentiation rules that the mixed
partial derivative I P- 11 can be written as the single determinant (indicated hereafter by
I (P -i)i = IP P -I ) generated by differentiating the rows of the matrix (P- ) in accor-
dance with r and then computing the determinant of the matrix derivative. That is, due to
the single-row dependence of (P- I) on each P(i I n), the two operations involved
(differentiating (P- ) and evaluating its determinant) commute. The same conclusion
applies to any mixed partial derivative of I P;- II with respect to the P(i I n)'s, and hence