Nondegenerate symmetric bilinear forms

Question

Let $K$ be field and $A$ is sub-algebra of $M_n(K)$. we define $b\colon A\times A \to K$ where $b(X,Y)=\operatorname{Tr}(XY)$. If $J(A )\not=0$, where $J(A)$ means Jacobson Radical of $A$, then $b$ is non-degenerate symmetric bilinear forms.

I can show $b$ is symmetric bilinear form, but I have no idea about non-degenerate.