Show any matrix $A_{n\times n}$ can be written as sum of two nonsingular matrices

Write $A=L+U$ where $L$ is lower triangular, $U$ is upper triangular. $L$ and $U$ have the same diagonal, equal to half the diagonal of $A$, except where zeros appear. In this case, use $1$ in $L$ and $-1$ in $U$. This makes the diagonals of $U$ and $L$ have no zero entries and so $L$ and $U$ are nonsingular.

Suppose that $\alpha\ne 0$ is not an eigenvalue of $A$. Then for any non-zero $v$ we have $(A-\alpha I)v\ne 0$, so $A-\alpha I$ is non-singular. So is $\alpha I$, and $A=(A-\alpha I)+\alpha I$.