# Interesting Matrix Question

1. Oct 8, 2004

### vsage

Are there two matrices A and B such that A*B is the zero matrix but B*A is not?

I'm leaning toward no.. I'm composing my solution right now.

Bah the only thing I can come up with is that if any row of A can be treated as a vector and any column of row B can be treated as a vector, for element (i, j) in the matrix AB will be 0 iff the vector of row i in A and column j in B are orthogonal (dot product is 0). I can't get much further right now :(

Last edited by a moderator: Oct 8, 2004
2. Oct 8, 2004

### TenaliRaman

A=
$$\left\{ \begin{array}{ccc} 1 & 1 \\ 0 & 0 \end{array} \right\}$$

B=
$$\left\{ \begin{array}{ccc} 1 & 1 \\ -1 & -1 \end{array} \right\}$$

-- AI

3. Oct 8, 2004

### vsage

Thanks. Although the question did just originally ask what is an example after many hours of scratching my head I made a proof that would satisfy that. Thank you for a template to go by though it facilitated the process a little.