They must be using the definition that 99% of all linear algebraists use: the matrix for D is the transpose of the coefficients matrix of the system: for example, you got , and since is the 3rd vector in the given matrix and the 2nd one , that means that the 3rd column of the matrix will have -1 in the 2nd position, which it indeed has.

BTW, the given matrix is the one for D wrt the given basis: the standard basis' vectors you use there have nothing to do with this problem since they don't belong to the space you're working with.

Tonio