How to inverse matrices?

The inverse of a matrix is one that when multiplied with the original matrix, forms an identity matrix. Rather than spell it out step by step, here is a link you will probably find useful. This is the method I use.

http://www.purplemath.com/modules/mtrxinvr.htm

Hope it helps.

In mathematics, Stirling numbers of the first kind, together with the Stirling numbers of the second kind, are one of the two types of Stirling numbers. They commonly occur in the study of combinatorics, where they count the number of permutations. The Stirling numbers of the first and second kind can be understood to be inverses of one-another, when taken as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind; further identities linking the two kinds, and general information, is given in the article on Stirling numbers. In mathematics, Stirling numbers of the first kind, together with the Stirling numbers of the second kind, are one of the two types of Stirling numbers. They commonly occur in the study of combinatorics, where they count the number of permutations. The Stirling numbers of the first and second kind can be understood to be inverses of one-another, when taken as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind; further identities linking the two kinds, and general information, is given in the article on Stirling numbers. Similar relationships involving the Stirling numbers hold for the Bernoulli polynomials. Many relations for the Stirling numbers shadow similar relations on the binomial coefficients. The study of these 'shadow relationships' is termed umbral calculus and culminates in the theory of Sheffer sequences.

For a general matrices, the process is quite involved and you have to know many concepts, row echelon, row operations, etc.
For a 2x2 and 3x3 matrix, there are published formulas for their inverses.
I cannot help you on that.