Least Angle Regression (LARS) without Matrix Inversion
If you want to solve for single value of $ \lambda $ in the model:

$$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1} $$

Then you can use Coordinate Descent method which is the fastest and simplest and doesn't require any matrix inversion.

I have a MATLAB code for in my $ {L}_{1} $ Regularized Least Squares Analysis GitHub Repository: