Expressing a matrice as a sum of two non singular matrices

[vish_maths]

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 ?

 

[micromass]

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

When is a triangular matrix non-singular??

 

[HallsofIvy]

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.