# Finding a General Form of Two non-zero, square Matrices with product zero

I am looking at two non-zero matrices, $A$ and $B$, both over $\mathbb{R}$ and of dimension $n \times n$. I think I've proven that their product $AB$ can be equal to zero by the following:

In the first entry of $AB=0$, which is $ab_{11}$ = $\sum_{i=1}^{n} a_{1n}b_{n1}$, there exists some $a_{1k}b_{k1} = -\sum_{i=1}^{n} a_{1n}b_{n1}$ for $i \neq k$. We just induce this for all remaining entries $ab_{ij}$ of $AB$.

Assuming that's a sufficient (and sufficiently elegant [I'm new]) proof, is there a general form of $A$ or $B$? My ultimate goal is to show that $BA \neq 0$ for all such $A$ and $B$, and I think knowing the general form of one of these matrices would help.

(Also, is it appropriate to ask more than one question about this same system on this same thread?)

This is a good way indeed to construct an example of matrices $A$ and $B$ such that $AB=0$. Still :

• this is not really elegant (my point of view, at least :))
• there is no better way in my own to construct the generic example of such $A$ and $B$.

One more ways to think of this problem : can you find an example of matrix $A$ such that $A^2=0$? or more generally $A^n=0$? This of course does not construct all examples of such $A$, $B$ but yet gives a nice sub-example (nilpotent matrices). Hint : look at matrice which are upper triangular, with $0$ on the diagonal.

The condition for $AB = 0$ is that all columns of $B$ are in the null space of $A$. So take $A$ to be any matrix of rank $< n$, find its null space, choose any $n$ vectors (not all $0$) in that null space, and let $B$ be the matrix whose columns are those vectors.

For all $(i,j)\in\{1,\cdots,n\}^2$, let define $E_{i,j}:=(\delta_{i,k}\delta_{j,\ell})_{k\in\{1,\cdots,n\}\\\ell\in\{1,\cdots,n\}}$, namely the square matrix of size $n$ with zeroes everywhere except at entry $(i,j)$. For all $(i,j),(k,\ell)\in\{1,\cdots,n\}^2$, notice that one has: $$E_{i,j}E_{k,\ell}=\delta_{j,k}E_{i,\ell}.$$ Therefore, if $j\neq k$, one has $E_{i,j}E_{k,l}=0$, without $E_{i,j}$ nor $E_{k,l}$ being zero.

Remark.

• Here $\delta_{i,j}$ is the Kronecker's symbol, it has value $1$ if and only if $i=j$ and $0$ otherwise.

• The matrices $E_{i,j}$ are called elementary has they form a basis for the space of all square matrices of size $n$ and are regularly involved in matrix problems, such that determining the set of square matrices commuting with all others.