Invertible matrix/Staircase form/Identity matrix/Fact

Let ${}K$ be a field, and let ${}M$ denote an invertible ${}n\times n$ -matrix over ${}K$ .

Then there exist

elementary row operations such that, after these manipulations, a matrix of the form

{\begin{pmatrix}d_{1}&\ast &\cdots &\cdots &\ast \\0&d_{2}&\ast &\cdots &\ast \\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1}&\ast \\0&\cdots &\cdots &0&d_{n}\end{pmatrix}}

with ${}d_{i}\neq 0$ arises. By further elementary row operation, one can also obtain the