# Writing a matrix as a sum of two invertible matrices

Let $n\geq 2$. Is it true that any $n\times n$ matrix with entries from a given ring (with identity) can be written as a sum of two invertible matrices with entries from the same ring ?

• You might want to consider the answer in the $n=1$ case as an example of how you could handle the general case. I've voted to close. – Ryan Budney Sep 5 '13 at 21:23
• Use upper triangular matrices with non zero diagonals – Alexandre Eremenko Sep 5 '13 at 22:05
• @Alexandre Eremenko: What if the ring is not a division ring? Additionally, over a field with two elements, the identity matrix can only be written as a sum of two invertible matrices as follows: $\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} = \begin{pmatrix}1 & 1 \\ 1 & 0\end{pmatrix} + \begin{pmatrix}0 & 1 \\ 1 & 1\end{pmatrix}$ – Ricardo Andrade Sep 5 '13 at 22:22
• I agree that my proposal does not work for all rings. – Alexandre Eremenko Sep 5 '13 at 22:28
• It is true for many rings. The terminology is somethig like "clean".There is also some USENET news posts on Sums of Invertible Matrices. A web search may help further. – The Masked Avenger Sep 6 '13 at 1:06

## 2 Answers

The answer is negative.

There is a nice theorem of M. Henriksen which says that If $n\geq 2$ then every element of $M_n(R)$ is a sum of three units also he proves that there are non-unit matrices in $\bf{M_2(\Bbb{Z}_2[x_1,x_2])}$ that can not be written as a sum of two units. You can find a copy of the article HERE

This may help. Any $2\times 2$ matrix is a sum of four units:

$$\begin{pmatrix} x & y \\ z & t \end{pmatrix} = \begin{pmatrix} x & 1 \\ -1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & -1 \\ 1 & t \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ z & -1 \end{pmatrix} + \begin{pmatrix} -1 & y \\ 0 & 1 \end{pmatrix}$$

• If e.g. z and y are each the sum of two units, you can reduce this to two summands. – The Masked Avenger Sep 6 '13 at 17:17
• +1 For the nice identity. – user38122 Sep 6 '13 at 18:47