Final answer:
To find the matrix of a quadratic form, we need to consider the coefficients of the squared terms as the diagonal elements and the half-coefficients of mixed terms as off-diagonal elements in a symmetric matrix. Two matrices corresponding to each quadratic form given in the question are provided.
Step-by-step explanation:
To find the matrix of a quadratic form, we express each term in the form of a matrix equation, such that the quadratic form equation can be written as xTAx, where X is the column matrix of variables, and A is the matrix we are looking for.
For the quadratic form 2x12 + 5x22 - 8x32 - 4x1x2 + 8x1x3 - 2x2x3, the matrix A is constructed by placing the coefficients of the xi2 terms on the diagonal and the coefficients of the xixj terms divided by 2 in the off-diagonal positions, respecting the symmetry of the matrix:
A =\[\begin{bmatrix}2 & -2 & 4 \\-2 & 5 & -1 \\4 & -1 & -8\end{bmatrix}\]
For the quadratic form 2x1x2 + 10x1x3 - 12x2x3, there are no squares of xi, hence the diagonal elements will be zero and again we divide the coefficients of xixj by 2 to place in the off-diagonals:
A =\[\begin{bmatrix}0 & 1 & 5 \\1 & 0 & -6 \\5 & -6 & 0\end{bmatrix}\]