Is the product of elementary matrices an elementary matrix?

I am reading about linear algebra and I have the following exercise:

Is the product of elementary matrices an elementary matrix?

I know that if $A$ is matrix, $(E_1 E_2 \dots E_n) A = A$. But How I can figure out that the elementary multiplications of matrices is an elementary matrix?

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Well, the only elementary matrices are (a) the identity matrix with one row multiplied by a scalar, (b) the identity matrix with two rows interchanged or (c) the identity matrix with one row added to another. Just write down any invertible matrix not of this form, e.g. any invertible $2\times 2$ matrix with no zeros. –  user15464 Sep 22 '12 at 15:32