# What is the difference between an identity matrix and reduced row echelon form?

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 is in reduced row echelon form. But there are many reduced row echelon matrices that are not an identity matrix, because they have zeros on the diagonal. It’s like saying “what’s the difference between a cat and a mammal?”: just because every cat is a mammal doesn’t make them the same concept. Commented Aug 29, 2019 at 17:18

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.

• I see, I didn’t consider how reduced row echelon form would function outside of a square matrix. Thanks! Commented Aug 29, 2019 at 17:35

One clear difference is that a reduced row echelon matrix can have zeroes on the main diagonal.

• Sorry for overlap with previous comment, It didn't appear on my screen until after I had answered.
– user502266
Commented Aug 29, 2019 at 17:21
• Oh ok, what is the use of having zeroes on the diagonal though, doesn’t that make it more difficult to solve simultaneous equations, or is it used for a different purpose? Commented Aug 29, 2019 at 17:29
• They may well arise when solving simultaneous equations - when various variables do not appear in a particular equation.
– user502266
Commented Aug 29, 2019 at 17:40