## 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 .

Remarks:

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 .)

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