The best way to find inverses is by Gaussian-Jordan elimination. Because that is the most efficient algorithm. However, it is not pleasant. There is a very elegant way to find inverses but when matrices get large it because computationally difficult to do (still it has important theoretical uses).

Step 1: Given a matrix compute . If this determinant is zero then it have no inverse.

Step 2: Compute thecofactor matrix. Meaning for each entry in the matrix compute the cofactor of . Replace each entry by its cofactor. This will form a matrix called the cofactor matrix.

Step 3: Compute theadjoint matrix. To do this find thetransposeof the cofactor matrix. The transpose operation on (denoted by ) is flipping the elements along the main diagnol. So for the entry replace by entry. (Note, that the main diagnol of the matrix is unchanged by transpose operation).

Step 4: Now divide each element in the adjoint matrix by the determinant. The resulting matrix is the inverse.