Building on our understanding of vectors and matrices, this chapter addresses the fundamental problem of solving systems of linear equations. Representing these systems in matrix form as $Ax = b$ is common in machine learning, particularly when determining model parameters like those in linear regression.

Here, we will examine methods to find the solution vector $x$. Key topics include:

• The concept and calculation of the matrix inverse ($A^{-1}$).
• Using the inverse to solve systems of the form $Ax = b$.
• The role of the determinant in assessing if a matrix is invertible.
• Computational approaches, including using NumPy to find inverses and solve systems, along with considerations for numerical stability.

We will also briefly review Gaussian elimination as a foundational technique for solving these systems. By the end of this chapter, you'll understand how to approach and solve linear systems that appear frequently in machine learning contexts.