For a function, replace (f(x)) with (y)
Swap (x) and (y)
Solve the new equation for (y)
Replace (y) with (f^{-1}(x))
Check that (f(f^{-1}(x))=x) and (f^{-1}(f(x))=x)
For a matrix, form the identity matrix beside it
Use row reduction to turn the original matrix into the identity matrix
The matrix on the other side becomes the inverse
Verify by multiplying the matrix and its inverse to get the identity matrix