Differentiation of A^21 (t). If a matrix A(t)and its inverse A^1 (t)are differen-
tiable with respect to t, then the derivative of A^1 (t)is given by
The derivative may be obtained by differentiating A(t)A^1 (t)with respect to t. Since
and
we obtain
or
dA-^1 (t)
dt
=-A-^1 (t)
dA(t)
dt
A-^1 (t)
A(t)
dA-^1 (t)
dt
=-
dA(t)
dt
A-^1 (t)
d
dt
[A(t)A-^1 (t)]=
d
dt
I= 0
d
dt
[A(t)A-^1 (t)]=
dA(t)
dt
A-^1 (t)+A(t)
dA-^1 (t)
dt
dA-^1 (t)
dt
=-A-^1 (t)
dA(t)
dt
A-^1 (t)
Appendix C / Vector-Matrix Algebra 881