### 30th Putnam 1969

**Problem B6**

M is a 3 x 2 matrix, N is a 2 x 3 matrix. MN =

8 2 -2
2 5 4
-2 4 5

Show that NM =
9 0
0 9

**Solution**

The key observation is that (MN)^{2} = 9 MN. [Of course, we expect this to be true since NM = 9 I, and it is easy to verify.]

It is also easy to check that MN has rank 2. The rank of NM must be at least as big as M(NM)N = 9 MN, so NM is non-singular. Now (NM)^{3} = N(MN)^{2}M = N(9 MN)M = 9 (NM)^{2}. Multiplying by the inverse of NM twice gives that NM = 9 I.

30th Putnam 1969

© John Scholes

jscholes@kalva.demon.co.uk

14 Jan 2002