Quiz, 2/14/2006, Math 221, s. 02
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[Note: the symbol l represents the letter "ell", not the numeral 1.
This was clear in the handwritten version of the quiz supplied in class.]
As a consultant to a top-secret government agency, you are informed
about two matrices A and B that A is k x n while B is m x l ,
AND that the product AB happens to be available; your job:
(i) draw some useful inference about k , l , m , and/or n .
Later research reveals that BA , too, is available.
(ii) Any further inference(s) to be drawn about k , l , m , n ?
(iii) Anything useful to be said about the size and shape of AB , BA ?
- - - - - - - -
(i) If k x n matrix A and m x l matrix B can be multiplied to form AB ,
then necessarily we must have had n = m (and AB is in fact a k x l matrix).
(ii) If B and A (as above) can be multiplied to form BA , then necessarily
we must have had k = l (and BA is in fact an m x n matrix).
(iii) When the situation is as described jointly by (i) and (ii), then both AB
and BA must be SQUARE matrices, with AB being k x k , BA being n x n .
Several "solutions" proposed that all four of k , l , m , and n must be
equal when both (i) and (ii) are valid. Alas, that just "ain't necessarily so."
(For example, A might be 2 x 3 , while B is 3 x 2 : then AB is
2 x 2 , and BA is 3 x 3 , but there's no reason to suspect 2 = 3 .)