Decomposing a matrix into elementary matrices [closed]

I am reading Milnor's Introduction to Algebraic K theory and come across the following claim on page 25: $\begin{bmatrix} A & 0 \\ 0 & A^{-1} \end{bmatrix}$ could be decomposed as a product of elementary matrices following from proof of 2.5, which contains an observation looking like $$ \begin{bmatrix} A & 0 \\ 0 & A^{-1} \end{bmatrix} = \begin{bmatrix} I & A \\ 0 & I \end{bmatrix} \begin{bmatrix} I & 0 \\ -A^{-1} & I \end{bmatrix} \begin{bmatrix} I & A \\ 0 & I \end{bmatrix} \begin{bmatrix} 0 & -I \\ I & 0 \end{bmatrix}.$$ Here an elementary matrix is a matrix that coincides with the identity except for a single off-diagonal entry. Any ideas on how to decompose such matrices into product of elementary matrices?

Solution 1:

The usual definition of elementary matrix is slightly different: for every elementary row transformation $\rho$ the elementary matrix $E(\rho)$ is the matrix obtained from the identity matrix $I$ by applying $\rho$. Milnor's elementary matrices correspond to $\rho$'s which add one row multiplied by a number to another row. If $\rho_1,..., \rho_m$ are elementary row transformations needed to transform $B$ to $I$ then $E(\rho_m^{-1})\cdot ...\cdot E(\rho_1^{-1})$ is a product of elementary matrices that is equal to $B$.

In your product of four matrixes $$\begin{bmatrix} I & A \\ 0 & I \end{bmatrix}\cdot \begin{bmatrix} I & 0 \\ -A^{-1} & I \end{bmatrix}\cdot \begin{bmatrix} I & A \\ 0 & I \end{bmatrix}\cdot \begin{bmatrix} 0 & -I \\ I & 0 \end{bmatrix} $$ the first three matrices require only Milnor's elementary matrices to reduce them to the identity matrix.

The fourth matrix also only needs these row transformations: First add rows $n+1$,...,$2n$ to rows $1,...,n$ respectively. Then subtract rows $1,...,n$ from rows $n+1,...,2n$.

Thus each of the four matrices is a product of "Milnor" elementary matrices and you are done.