I have a question and i hope that i will find the answer her,
So we all know that the reduced row echelon form for a invertible matrix is the matrix $I_n$, So what about the reduced row echelon form for a non invertible matrix, i want basically to know if there is a specific form for the last one.
Thank you in advance.
Best Answer
RREF of any $n\times n$ non-invertible matrix (with dimension of row or column space $m$) has the form as
$$\begin{bmatrix}1 & 0 &0 &\cdots&0&* &\cdots&*&*\\ 0&1&0&\cdots&0&* &\cdots&*&*\\ 0&0&1 &\cdots&0&* &\cdots&*&*\\ \vdots &\vdots &\vdots &\ddots & \vdots &\vdots &\vdots &\vdots &\vdots\\ 0&0&0&\cdots&1 &*&\cdots&*&* \\0 &0&0 &\cdots &0 &0 &\cdots &0&0\\\vdots &\vdots &\vdots &\ddots & \vdots &\vdots &\vdots &\vdots &\vdots \\0 &0&0 &\cdots &0 &0 &\cdots &0&0 \end{bmatrix} $$
where this matrix has $m$ non zero rows and $*$ can be any number.
This form is modified RREF. We can get this form after arranging columns of the original RREF.