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Unformatted text preview: of the matrices as follows, an
a 2l a 22 a 2n X2 (B.61)
Yn a nI . .. a n2 We c an solve this set of equations for X l, X 2, • •.
using Cramer's rule [see Eq. (B.31)]. This yields
the matrices are n o longer conformable for t he product. I t is evident t hat in general, Xl A B =F B A X2 Indeed, A B m ay exist and B A may not exist, or vice versa, as in t he above examples. We shall see later t hat for some special matrices,
A B==BA ( A+B)C == A C+BC (B.58) C (A + B ) == C A + C B (B.59) We can verify t hat any matrix A premultiplied or postmultiplied by the identity
matrix I remains unchanged: in terms of Y l, AI
IfAil Xn IT.q1 lilT I llifL Y 2, . .. , Y n by Yl
Y2 (B.62) . ...................... (B.57) When Eq. (B.57) is true, matrices A and B are said to c ommute. We m ust stress
here again t hat i n general, matrices do not commute. Operation (B.57) is valid only
for some special cases.
In the m atrix p roduct A B, m atrix A is said to be p ost m ultiplied by B or
matrix B is s aid to be p remultiplied by A . We may also verify t he following
relationships: Ifl,1 Ill,1
Illr l Illr l Xn a nn
, Xn Yn in which IAI is t he d eterminant of t he m atrix A a nd IDij I is t he cofactor of element
in t he m atrix A . T he cofactor of element a ij is given by ( -1 ) i+j t imes t he
d eterminant of t he (n - 1) x (n - 1) m atrix t hat is obtained when t he i th row a nd
t he j th column in matrix A are deleted.
We can express Eq. (B.61) in matrix form as a ij y ==Ax (B.63) We can now define A -I, t he inverse of a square matrix A , w ith t he p roperty
Then, premultiplying b oth sides of Eq. (B.63) by A -I, we o btain (B.64) 40 B ackground B .6-3 or
X= A - 1y Derivatives and Integrals o f a Matrix (B.65)
Elements o f a m atrix need not b e c onstants; t hey may be functions of a variable:
For example, if A comparison of E q. (B.65) with Eq. (B.62) shows t hat IDlll ID211 IDnll I Dd ID221 A -I = _1_ IDn21 IAI
IDlnl ID2nl sin t
(B.66) IDnnl (B.67a)
Postmultiplying t his e quation by A -I a nd t hen premultiplying by A , we c an show
t hat A A- 1 = I (B.68) t hen t he m atrix elements are functions of t. Here, i t is helpful t o d enote A by A (t).
Also, it would be helpful t o define t he derivative a nd integral of A (t).
T he derivative of a m atrix A (t) ( with respect t o t ) is defined as a matrix whose
i jth element is t he derivative (with respect t o t) of t he i jth element of t he m atrix
A . T hus, if A (t) = [ aij(t)]mxn
t hen ~[A(t)] = [~aij(t)]
Thus, t he derivative of t he m atrix in Eq. (B.68) is given by (B.67b) A (t) = - [ E xample B .12 (B.69a)
m Xn or Note t hat t he m atrices A a nd A -I commute. 2 - 2t
e cos t
2e - 2t
- t -e -t Similarly, we define t he i ntegral of A (t) ( with respect t o t) as a m atrix whose
i jth element is t he integral (with respect t o t) of t he i jth e lement of t he m atrix A: L et us f ind A - 1 if j A (t) dt = (j (B.70) a ij(t) d t) m xn T hus, for t he m atrix A in Eq. (B.68), we have Here = - 4,
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