Question

The Nature of quadratic form Q(x) = x² + 3y² +3z² is 

Answer 1

Matrix associated to this quadratic form $Q(x)=x^{2}+3y^{2}+3z^{2}$ is

$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}$

Note that the matrix is symmetric. As each principle minors of this matrix has their determinant positive (1, 3 and 9 respectively), we can conclude that the matrix is positive definite and therefore the associated quadratic form is positive definite.