From inverse formula of a matrix

A⁻¹ = (1/(det(A)))adj(A).

If the matrix A is singular, that is  det(A) = 0, then we can not find the inverse of A.

But for every finite matrix A (square or rectangular) of real or complex elements, there is a unique generalized inverse, namely, Moore-Penrose invere  X  satisfying the four equations

AXA = A;  XAX = X;  (AX)* = AX;  (XA)* = XA;

where * is means conjugate transpose. See 

A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory and Applications, Second Edition, Springer-Verlag, New York, 2003.  pp.40-50.

There are another generalized inverse, for examples "group inverse of a square matrix" and "Drazin inverse of a square matrix."

Remark: if a square matrix  A  is nonsingular then 

A⁻¹ = A+  (the Moore-Penrose inverse) = A#  (the group inverse).

As said Prof. Wiwat in the answers above. Just I add the  Gauss-Jordan echelon form method for the non singular matrix, you perform it on the extended matrix ( A  In ), you obtain ( In  A-1 ). But still the inverse formula of a non singular matrix the most popular. If you don't interested in knowledge and want just results ( calculation) you can use some software of mathematics, the most popular and easy is Scientific work place, just write your matrix A -1 and click on the sign =? in the toolbar.

For a singular matrix, we can find inverse (pseudo inverse) using pinv() command in matlab. we can find inverse not only for singular matrices but for non-square matrices also using pseudo inverse concept.