Inverses of linear transformations

A linear transformation, in essence, is a function from one vector space to another that preserves the operations of addition and scalar multiplication.
Inverses of linear transformations, also known as inverse transformations, are transformations that can ‘undo’ the transformations applied by original transformations.
If ‘T’ is a linear transformation from a vector space ‘V’ to another vector space ‘W’, then an inverse transformation ‘S’ exists if ‘S’ is also a linear transformation from ‘W’ back to ‘V’ such that when you apply ‘T’ first and then ‘S’, or ‘S’ first and then ‘T’, you get back the original vector.
This can be written as ST = TS = I, where ‘I’ is the identity transformation, which leaves each vector unchanged.
A transformation T will only have an inverse if it is both one-to-one (no two vectors in ‘V’ map to the same vector in ‘W’) and onto (for every vector in ‘W’, there is at least one vector in ‘V’ that maps to it).
In terms of matrices, inverse transformations correspond to inverse matrices. If ‘T’ is represented by a matrix ‘A’, ‘S’ is represented by the inverse matrix ‘A^-1’.
To determine if the inverse of a linear transformation exists, we can use the determinant of the matrix. If the determinant of the matrix ‘A’ is not zero, then ‘A’ has an inverse.
If the inverse exists, it can be calculated by adjoint of ‘A’ divided by the determinant of ‘A’ where the adjoint is the transpose of cofactor matrix.