I have been learning about matrices recently and have come across the terms reduced row echelon form and identity matrix. At first glance, they seem to be identical - a row of ones on the diagonal, with the other entries being zero.
My question is whether there is a difference between reduced row echelon form and an identity matrix?
An identity matrix must be square, but that's not required for reduced row echelon form. All of these $2\times 4$ matrices are in RREF: $$ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\qquad \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}\qquad \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} $$ Also, keep in mind that a matrix can have entries other than zeroes and ones and still be in RREF. For instance: $$ \begin{bmatrix} 1 & 0 & 2 & 0 & 3 \\ 0 & 1 & 4 & 0 & 5 \\ 0 & 0 & 0 & 1 & 6 \end{bmatrix} $$ is in RREF.
You might find the Wikipedia article on REF useful.
One clear difference is that a reduced row echelon matrix can have zeroes on the main diagonal.
