Step 1: Given a matrix compute . If this determinant is zero then it have no inverse.
Step 2: Compute the cofactor 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 the adjoint matrix. To do this find the transpose of 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.