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 \(\mathbb K=\mathbb Z_3\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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