1.2 #T4:

PROBLEM:
Let k be a real number, let A be an mxn matrix, and let 
O be the mxn matrix of all zeros. If it is given that kA=O
prove that either k=0 or A=O.

SOLUTION:
Suppose A is not equal to O. Then at least one entry of A is not
zero. Let a(ij) be a nonzero entry in A. (I cannot type subscripts,
so I am using a(ij) instead.) Then k * a(ij) = 0. This implies k=0 
since we can divide through by a(ij). 

Now suppose k is not 0. Let a(ij) be any entry of A. Then  k * a(ij) = 0,
since kA=O is given. But we can divide through by k and get a(ij) = 0.
But a(ij) could have been any entry of A. Thus, if k is not 0, all 
the entries of A must be zero and A=O.

[Compare with axioms problem 9.]