The inverse of a given square matrix, $A$, is the matrix $A^{-1}$ such that $AA^{-1}$ is the identity matrix.
In theory, matrices need to be inverted to solve systems of linear equations. Inverse matrices play an important role in theoretical analyses.
In practice, you never want to explicitly invert a matrix. It is computationally expensive and often numerically unstable. Alternatives exist for all common uses of matrix inverses, such as Gaussian elimination for solving linear equations.
