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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?)

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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.

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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.

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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.

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