Propositions of elementary matrix

Question

i'm trying to solve a question about elementary matrix.

When given $A_{m,n}$ and $B_{n,p}$ which differ from the Zero matrix.

Also, multiplying of $A$ and $B$ is the zero matrix, that is: $AB=0$;

1. If $C$ is an elementary matrix of $A$ => $CB=0$?
2. If $C$ is an elementary matrix of $B$ => $AC=0$?

My intuition feelings are that one is true, and two is false.

I would like your help.

Thanks a lot!

Answer 1

Suppose $C$ is obtained from $A$ by applying an elementary row operation. This is equivalent to saying that $C=EA$, where $E$ is a suitable invertible matrix¹.

Then $CB=EAB=E0=0$.

On the other hand, if $C=EB$ is obtained from $B$ by an elementary row operation, it's not necessarily true that $AC=0$. Example: $$ A=\begin{bmatrix}0&1\\0&0\end{bmatrix}, \qquad B=\begin{bmatrix}1&0\\0&0\end{bmatrix} $$ Then $AB=0$, but if $C$ is obtained from $B$ by swapping the rows, we have $$ AC= \begin{bmatrix}0&1\\0&0\end{bmatrix} \begin{bmatrix}0&0\\1&0\end{bmatrix} = \begin{bmatrix}1&0\\0&0\end{bmatrix} $$

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Footnotes.

¹ The matrix $E$ is obtained by applying the same elementary row operation on the identity matrix.

Answer 2

See here for a nice intro to Elementary Matrixes.

First, define $I_R$ as the matrix that results from a single elementary operation on the rows of the appropriately sized identity matrix $I$.

Then $C_R = I_R \times A$ is an elementary matrix of A. It follows then that $$C_R \times B = (I_R \times A) \times B$$ $$= I_R \times (A \times B)$$ $$I_R \times 0$$ $$=0.$$

Now define the elementary matrix $C_C = B \times I_C$ where $I_C$ is the matrix that results from a single elementary operation on the columns of the appropriately sized identity matrix $I$. Then

$$A \times C_C = A \times (B \times I_C)$$ $$= (A \times B) \times I_C$$ $$0 \times I_C$$ $$=0.$$

Both assertions can be true if the construction of the elementary matrix is done appropriately (in this case a row operation for A and a column operation for B.) However, as is pointed out, one of the assertions can fail if the elementary matrix is constructed as a row operation or as a column operation for both A and B.