Symmetric matrix of a form

Task number: 2738

Let

$$\mathbf A=\begin{pmatrix} 1 & -2 & 0 \\ 2 & 0 & -1 \\ -2 & 2 & 0 \\ \end{pmatrix}$$

be the matrix of a bilinear form on $$\mathbb K^3$$. Determine the analytic expression of this form and also for the associated quadratic form. Find a symmetric matrix for this quadratic form. (All tasks w.r.t. the same basis.)

• Variant 1

For $$\mathbb K=\mathbb R$$.

• Variant 2

For $$\mathbb K=\mathbb Z_2$$ (the number 2 in $$\mathbb Z_2$$ interpret as 0).

• Variant 3

For $$\mathbb K=\mathbb Z_3$$.