158 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5- Let U = (1,2,3). Consider the assertion that for all subsets A, B, and C of U,
(A A B) A C = A A (B A C); that is, the operation of symmetric difference is associa-
tive over this particular universal set U.
(a) How many particular cases are encompassed by this statement for the given
universal set?
(b) Verify that the statement is true for three such cases.
(c) Do you believe that symmetric difference is associative in general? How would
you go about investigating this possibility? - An m x n real matrix is a rectangular array of m rows and n columns of real
numbers, called entries. The entry in the ith row, jth column is customarily denoted
aij and A itself may be represented
which may, in turn, be abbreviated A = (aij), , ,. A is said to be square if m = n.
Some readers are probably familiar with elementary matrix theory, but for those
who may not be, we provide here some basic definitions that we use in the following
exercises and later in this chapter.- If A = (aij),, and B = (bij),. ., define A = B if and only if aij = bij for all i =
1,2 ,..., m,andj=1,2 ,..., n. - If A = (aij),. and B = (bij),. ., define A + B by the rule A + B = (aij + bij), , ..
If k E R, define kA = (kaij) ,... - If A = (aij), , , and B = (bijln , p, define AB by the rule AB = (cij), , p, where
Cij = xi=laikbkj. - If A = (aij), , ,, define A', the transpose of A, by A' = (bij), , ,, where bij = aji
for all i = 1,2,... , n, and j = 1,2,... , m. - A square matrix A = (aij),. , is said to be symmetric if and only if A' = A and
antisymmetric if and only if A' = -A. - If A = (aij),,,, we define the determinant of A, denoted IAl, by the rule
IAl = a1 ,a22 - a12a21-
Throughout parts (a) through (h), assume and use the facts that when the appro-
priate quantities are defined, matrix addition and multiplication are associative,
matrix addition is commutative, and matrix multiplication distributes over matrix
addition. Note that matrix multiplication is not commutative, even when both AB
and BA are defined. Assume also that if A + B and AB are defined, then
(A + B)' = A' + B', (AB)' = B'A' and (kA)' = kAt for any k E R.
(a) Prove that if A and B are m x n matrices with A' = B', then A = B.
(b) Prove that (At)' = A for any matrix A.
(c) Prove that if A, B, and C are matrices of the same shape, then (A + B + C)' =
At + Bt + C'.
(d) Prove that if the product ABC is defined, then the product C'B'At is defined
and CtB'At = (ABC)'.
(e) Prove that if A and B are symmetric square matrices, then A + B is symmetric.