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