> 
> Let $A$ be a square matrix of order $3$. Prove that 
>  $$
> \operatorname{adj}(A) 
> = \tfrac{1}{2} \bigl[ 
> (\operatorname{tr} A)^2 - \operatorname{tr}(A^2) \bigr] I\_3 
> - [\operatorname{tr} A] A + A^2
> $$ 
>  where $\operatorname{tr}A$ is the trace of $A$.
> 
> 
> 

To begin, I'm not quite sure how to start proving this. I've considered using brute force, but I suspect there should be a much more elegant way of doing it. Is there a need to prove this for both $(i,i)$ and $(i,j)$ entries? I'd appreciate an explanation that is not too complex. Thanks in advance!