You had the right answer!

But as explanation, lets say A is your original matrix:

The definitions are best explained backwards:

. the adjunct adj(A) equals the transpose of the cofactor matrix C

. the transpose of a matrix is one with the rows and columns flipped

. The cofactor matrix C is the matrix of minors M, with each position multiplied by its respective sign

. For a 2 x 2 matrix the minor of each position M[r,c] is the diagonally opposite element.

Using these definitions and starting with A, first create the matrix of minors:

Then the cofactor matrix C has each element of M multiplied by its sign where r,c are the row and column number of each position.

Finally so:

The calculation of adj for 3x3 and larger matrices is the same except that the calculation of the minor matrix requires calculating determinants.