Is $\left<ad-bc,a+d\right>\subseteq k[a,b,c,d]$ a reduced ideal, and if yes, why?

Question

Is $\left<ad-bc,a+d\right>\subseteq k[a,b,c,d]$ a reduced ideal, and if yes, why?

I have tried to prove it, but I have so far failed. The locus of this ideal can be identified with the $2\times 2$ matrices with only $0$s as eigenvalues.