Which of the following real quadratic form on $$\R^2$$ is positive definite .​

Correct option is B

Result : If all the leading principle minors of the matrix of a quadratic form are positive,

then it is a positive definite matrix .  

Solution :

​$$\text{The matrix of the quadratic form} \\Q(X, Y) = X^2 - XY + Y^2 \ is \\\begin{bmatrix}1 & -\frac{1}{2} \\-\frac{1}{2} & 1\end{bmatrix},\\\text{whose leading principal minors are:}\\ \Delta_1 = \det \begin{bmatrix} 1 \end{bmatrix} = 1\text{ and } \Delta_2 = \det \begin{bmatrix}1 & -\frac{1}{2} \\-\frac{1}{2} & 1\end{bmatrix} = \frac{3}{4}.\\\text{Both of them are positive}\\\textbf{ Hence, it is positive definite.}$$​​