Is there any Trick to compute the Adjoint of a $3 \times 3$ Matrix?

Question

I want to know if there exists somekind of trick to calculate the Adjoint of a $3 \times 3$ Matrix, just like we calculate the Adjoint of a $2 \times 2$ Matrix

Answer 1

It is an easy exercise to check that the adjoint of an $n\times n$ matrix $A$ is the conjugate transpose. That is, if $A$ has entries $A_{kj}$, then $A^*$ has entries $\overline{A_{jk}}$.

This is very easy to check: $$ (A^*)_{kj}=\langle A^*e_j,e_k\rangle=\langle e_j,Ae_k\rangle =\overline{\langle Ae_k,e_j\rangle}=\overline{A_{jk}}. $$