Answer
False
Work Step by Step
Suppose $A\begin{bmatrix}
3&0\\0&1
\end{bmatrix}$ and $B=\begin{bmatrix}
1 & 0\\0 &3
\end{bmatrix}$ are elementary matrices
We can see that their product $\begin{bmatrix}
3&0 \\0 &3
\end{bmatrix}$ is not an elentary matrix.
Hence, the statement is false.
