# Adding identity to invert matrix

I'm looking into algorithm implementation which is essentially linear regression:

$$\|Ax - b\| \rightarrow \min$$

Matrix A and vector b are estimated using data, then we do $$A^{-1}b$$ to find x. But prior to taking the inverse, there is a check if $$det(A)$$ less than 1, if it is, then we do $$A + I*\epsilon A$$ until $$\det(A)$$ reaches 1.

What could be the reason for requiring $$\det(A)$$ to be 1 or bigger to safely invert a matrix. The determinant is not condition number, does it "estimate" "distance to singularity".

Update 1 (adding more context):

x is a vector which is reshaped into an image filter and used for image filtering. Does this process make the matrix "more" unitary?

• Is there any more context? Because indeed, $\det(A)$ can be 1 and still the matrix can be arbitrarily ill-conditioned at the same time. – Algebraic Pavel Jul 9 at 14:58
• @AlgebraicPavel added – Fortyq Jul 9 at 19:02
• It might have to do with underflow. – EuxhenH Jul 9 at 19:16
• (1) are you adding $\epsilon A$ or $\epsilon I$ to $A$? I.e., why is the $I$ there if it's the former?, (2) it could be viewed as a regularization of the problem (for when it is close to being ill-defined especially but also just so the output is nicer, e.g. smoother, or smaller coefs), (3) can you post more context? a paper? – user3658307 Jul 10 at 1:23