# Expressing a matrice as a sum of two non singular matrices

Hello everyone , So here is this problem which i was recently thinking about
Expressing any matrix as the sum of two non singular matrices
So, when i think of ways to express a matrix as sum of two matrices, the thought which
comes first is :

(a) Any matrix can be expressed as the sum of a symmetric and an asymmetric matrix
but an asymmetric matrix is always singular which means this option is ruled out

(b) Suppose A and B are two non Singular matrices. There ought to be some technique
of factorising A and B so that some common terms exist and when i combine these two, a term is obtained whose property defines whether the resulting matrix is singular or not

What can be such a factorisation ?

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## Answers and Replies

Perhaps express it as a sum of an upper-triangular and a lower-triangular matrix??

When is a triangular matrix non-singular??

HallsofIvy
Science Advisor
Homework Helper
but an asymmetric matrix is always singular which means this option is ruled out

Where did you get that idea? First, "asymmetric" simply means "not symmetric"- you mean "anti-symmetric". And even for anti-symmetric matrices, this is not true:
$$\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}$$
is anti-symmetric but has determinant 1 and so is not singular.