Let A = (a$$_{i,j}$$​ ) be a real symmetric 3 × 3 matrix. Consider the quadratic form

$$Q(x_1, x_2, x_3) = \mathbf{x}^T A \mathbf{x}, \quad \text{where } \mathbf{x} = (x_1, x_2, x_3)^T$$​.

Which of the following is true?

Correct option is B

​$$\textbf{Let} \quad A = \begin{bmatrix} a & d & e \\ d & b & f \\ e & f & c \end{bmatrix}, \quad \text{and suppose} \quad Q(x_1, x_2, x_3) \text{ is positive definite.}\\[10pt]\textbf{Positive Definite Condition:} \\[10pt]\text{For } Q(x_1, x_2, x_3) \text{ to be positive definite:} \\[10pt]\text{1. All principal minors of } A \text{ must be positive:} \\D_1 = [a], \quad D_2 = \begin{bmatrix} a & d \\ d & b \end{bmatrix}, \quad D_3 = \begin{bmatrix} a & d & e \\ d & b & f \\ e & f & c \end{bmatrix}.\\\\[10pt]\text{2. Calculating the determinants:} \\D_1 = a > 0.\\D_2 = \det\begin{bmatrix} a & d \\ d & b \end{bmatrix} = ab - d^2 > 0 \implies ab > d^2.\\D_3 = \det(A) = abc + 2def - af^2 - be^2 - cd^2 > 0.\\[10pt]\textbf{Verification of Options:} \\[10pt]\textbf{Option A:} \\[10pt]\text{If } Q(x_1, x_2, x_3) \text{ is positive definite, this does not imply that } \\a_{i,j} > 0 \text{ for } i \neq j.\\ \text{For example, consider } \\A = \begin{bmatrix} 4 & 2 & -3 \\ 2 & 5 & 1 \\ -3 & 1 & 6 \end{bmatrix}:abc + 2def = 108, \quad af^2 + be^2 + cd^2 = 73, \quad 108 > 73,\\\text{but } e = -3 < 0. \\\text{ Thus, } a_{i,j} > 0 \text{ for } i \neq j \text{ is not necessary for } Q \text{ to be positive definite.} \\\textbf{Option A is incorrect.}\\[10pt]\textbf{Option B:} \\[10pt]\text{If } Q(x_1, x_2, x_3) \text{ is positive definite, then all diagonal elements } a_{i,i} > 0. \\\text{From the positivity of principal minors:}\\ D_1 = a > 0, \quad D_2 = ab - d^2 > 0 \implies b > 0, \quad D_3 > 0 \implies c > 0.\\\text{Hence, } a_{i,i} > 0 \text{ for all } i.\\ \textbf{Option B is correct.}\\[10pt]\textbf{Option C:} \\[10pt]\text{If } a_{i,j} > 0 \text{ for } i \neq j, \text{ this does not guarantee that }\\ Q(x_1, x_2, x_3) \text{ is positive definite.} \\\text{For example, if }\\ A \text{ has positive off-diagonal entries but fails to satisfy the positivity of principal minors, } \\Q \text{ will not be positive definite.}\\ \textbf{Option C is incorrect.}\\[10pt]\textbf{Option D:} \\[10pt]\text{If } a_{i,i} > 0 \text{ for all } i, \text{ this does not guarantee that } \\Q(x_1, x_2, x_3) \text{ is positive definite.} \text{For instance, if }\\ A = \begin{bmatrix} 4 & 6 & 7 \\ 6 & 5 & 1 \\ 7 & 1 & 6 \end{bmatrix},\text{where } a, b, c > 0,\\ \text{ but if the determinant } D_3 \leq 0, Q \text{ will not be positive definite.}\\ \textbf{Option D is incorrect.}\\[10pt]\textbf{Final Answer:} \\\textbf{Option B is correct.}$$​​