Proof: Note that we compute the (i, j)th entry of the matrix A1- 'A by the
formula
(Ai-A1)i,,j = -(Aj-1)i,k(Ai)kj
k=1
= (a-1),jjajj, + (a-1)i,jaj,j + (a-1)i,j+laj+,j,
where (a-1)i,j denotes the (i, j)th entry of the matrix A1-'. For j = i, we have
(Ai- Al),,i = [(i 1)(n i + 1)(-1) + i(n i + 1)(2) + i(n i)(-1)] = 1.
For j 7 i, we consider two cases:
1
(A1-1A)i,j = 1 [(j-1)(n-i+1)(-1)+j(n-i+1)(2)+(j+1)(n-i+1)(-1)] = 0,
ifj= 1,2,.---. ,i-1, and
1
(Ail-1A)i', = -- [i(n-i-(m-1))(-1)+i(n-i-m)(2)+i(n-i-(m+1))(-1)] = 0,
if j = i + m, where m = 1, 2, ,n i. Therefore, A-'A = I.
Remark: Dr. J. Gopalakrishnan suggests to use the finite element method to
find Ai1 Consider the two-point value problem:
U"= f on (0, 1),
U() = U(1) =0.
Then the Green's function G(x, s) is given by
G(x, (1 s)x if x < s,
(1 x)s if x > s.
Let 0 = Xo < xi < -.- < Xn+i = 1 with xi+1 xa = 1/(n + 1) for all i = 0,1,...,n.
Define
a(u, v) = u'(x)v'(x)dx,