Math 222 - Sample Proof Solutions
Instructor - Al Boggess
Spring 1998
Section 3 - Page 55
25. We are given that AT = ,A and we want to show that aii = 0 (the
diagonal entries). The ij th entry of AT is aji, which is assumed to
equal ,aij . So
aji = ,aji
Setting i = j gives aii = ,aii , which can only happen if aii = 0.
26. a) We are given that B = A + AT and C = A , AT , and we are to
show that B T = B and C T = ,C . We have
B T = (A + AT )T = AT + (AT )T = AT + A = B
So B T = B . We also have
C T = (A , AT )T = AT , (AT )T = AT , A = ,(A , AT ) = ,C
as desired.
b) We are to show that a matrix A can be written as the sum of a
symmetric and skew-symmetric matrices. Note that in part a), B is
symmetric and C is skew-symmetric. We have
B + C = (A + AT ) + (A , AT ) = 2A
or
B + C = A + AT + A , AT = A
2 2
2
2
Since B is symmetric, so is B=2. Likewise C=2 is skew-symmetric.
Therefore A is the sum of a symmetric matrix, B=2, and a skewsymmetric matrix C=2, as desired.
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