Can AB be zero while BA is nonzero for matrices A and B?

[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 :(

 

[TenaliRaman]

A=
$$\left\{ <br /> \begin{array}{ccc} <br /> 1 & 1 \\ <br /> 0 & 0 <br /> \end{array} <br /> \right\}$$

B=
$$\left\{ <br /> \begin{array}{ccc} <br /> 1 & 1 \\ <br /> -1 & -1 <br /> \end{array} <br /> \right\}$$

-- AI

 

[vsage]

A=
$$\left\{ <br /> \begin{array}{ccc} <br /> 1 & 1 \\ <br /> 0 & 0 <br /> \end{array} <br /> \right\}$$

B=
$$\left\{ <br /> \begin{array}{ccc} <br /> 1 & 1 \\ <br /> -1 & -1 <br /> \end{array} <br /> \right\}$$

-- AI

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.