To determine which of the given matrices is a matrix of positive quadratic form, we need to check if the matrix is positive definite. A matrix is positive definite if all its leading principal minors are positive.

The matrices in the question are:

1. Matrix (a): $\begin{pmatrix} 2 & 3 \\ 3 & 11 \end{pmatrix}$
2. Matrix (b): $\begin{pmatrix} 2 & 3 \\ 3 & 3 \end{pmatrix}$
3. Matrix (c): $\begin{pmatrix} 2 & 3 \\ 4 & 11 \end{pmatrix}$
4. Matrix (d): $\begin{pmatrix} 2 & 3 \\ -3 & 11 \end{pmatrix}$

Steps to determine positive definiteness:

1. For a 2x2 matrix $\begin{pmatrix} a & b \\ b & d \end{pmatrix}$, it is positive definite if:
   • $a > 0$
   • $\text{det}(A) = ad - b^2 > 0$

Let's evaluate each option based on these criteria:

Matrix (a):

• $a = 2 > 0$
• Determinant: $2 \cdot 11 - 3^2 = 22 - 9 = 13 > 0$
• Conclusion: Matrix (a) is positive definite.

Matrix (b):

• $a = 2 > 0$
• Determinant: $2 \cdot 3 - 3^2 = 6 - 9 = -3$, which is not greater than 0.
• Conclusion: Matrix (b) is not positive definite.

Matrix (c):

• $a = 2 > 0$
• Determinant: $2 \cdot 11 - 3 \cdot 4 = 22 - 12 = 10 > 0$
• Conclusion: Matrix (c) is positive definite.

Matrix (d):

• $a = 2 > 0$
• Determinant: $2 \cdot 11 - 3 \cdot (-3) = 22 + 9 = 31 > 0$
• Conclusion: Matrix (d) is positive definite.

Answer

Matrices (a), (c), and (d) satisfy the positive definiteness condition. However, the question may intend only one answer, in which case, option (a) is the most straightforward positive definite matrix without complex entries or signs.

Would you like to delve deeper into positive definite matrices, or do you have any other questions?

1. How do leading principal minors relate to matrix positivity?
2. What’s the practical significance of positive definite matrices in applications?
3. Could these matrices be used in quadratic optimization problems?
4. Why does matrix (b) fail to be positive definite specifically?
5. Are there methods to confirm positive definiteness for larger matrices?

Tip: For 2x2 matrices, verifying positive definiteness is simple—just check the top-left entry and the determinant.