Matrix division can be expressed as either A/B or B\A, both of which can be computed without directly finding the inverse of B, provided B is not singular. The equations X = A/B and X = B\A can be solved using Gaussian elimination, yielding unique solutions if B is non-singular. In cases where B is singular, the pseudo-inverse must be used, often calculated through singular value decomposition. Practical applications of matrix division include solving systems of linear equations, optimization problems, and data fitting in various fields such as statistics and machine learning. Understanding these methods enhances the ability to work with complex matrix operations in real-world scenarios.