M is a 3 x 2 matrix, N is a 2 x 3 matrix. MN =
8 2 -2 2 5 4 -2 4 5Show 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)2M = N(9 MN)M = 9 (NM)2. Multiplying by the inverse of NM twice gives that NM = 9 I.
© John Scholes
jscholes@kalva.demon.co.uk
14 Jan 2002