This implies that the matrix product ZX =C is a constant. If the matrix X is
nonsingular, then X−^1 exists, so that one can solve for Z as Z=X−^1 C. At time
t= 0 , it is required that Z(0) = Iand X(0) = Iso that C=I and therefore Z=X−^1.
Consequently, the adjoint equation can be expressed in the form
dX −^1
dt
=−X−^1 A(t), X −^1 (0) = I (10 .15)
Now multiply equation (10.9) on the left by X−^1 to obtain
X−^1 d ̄y
dt
=X−^1 Ay ̄+X−^1 f ̄(t) (10 .16)
and then multiply equation (10.15) on the right by y ̄to obtain
dX −^1
dt
y ̄=−X−^1 Ay ̄ (10 .17)
Sum the equations (10.16) and (10.17) to obtain
X−^1 ddt ̄y+dX
− 1
dt y ̄=
d
dt
(
X−^1 y ̄
)
=X−^1 f ̄(t) (10 .18)
Integrate equation (10.18) from 0 to tand show
∫t
0
d
dt
(
X−^1 (t) ̄y(t)
)
dt =
∫t
0
X−^1 (t)f ̄(t)dt
which produces the result
X−^1 (t) ̄y(t)
t
0
=X−^1 (t) ̄y(t)−X−^1 (0) ̄y(0) =
∫t
0
X−^1 (t)f ̄(t)dt
which indicates that the solution to the matrix equation (10.9) can be represented
in the form
̄y(t) = X(t) ̄c+X(t)
∫t
0
X−^1 (ξ)f ̄(ξ)dξ (10 .19)
The Determinant of a Square Matrix
Afundamental principle from probability and statistics is that if something can
be done in ndifferent ways and after it has been done in one of these ways, a second
something can be done in mdifferent ways, then the two somethings can be done
in the order stated in n·mdifferent ways. If a third something can be done in p
different ways, then the three somethings can be done in n·m·pdifferent ways. This
