Math Problem Statement
Obtain the nature of the quadratic form 2x² + 3y² + 22 + 2xy
Solution
To determine the nature of the quadratic form , we need to analyze the form by examining its associated matrix and finding its discriminant.
Step 1: Write the Quadratic Form in Matrix Notation
The general quadratic form in two variables, , can be expressed in matrix form as:
\begin{bmatrix} a & h \\ h & b \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$ where: - $$a = 2$$ (coefficient of $$x^2$$), - $$b = 3$$ (coefficient of $$y^2$$), - $$h = 1$$ (half of the coefficient of $$xy$$, which is 2). The corresponding matrix for this quadratic form is: $$\mathbf{A} = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}$$ ### Step 2: Calculate the Determinant of the Matrix The discriminant (or determinant) of matrix $$\mathbf{A}$$ is given by: $$\det(\mathbf{A}) = ab - h^2 = (2)(3) - (1)^2 = 6 - 1 = 5$$ ### Step 3: Determine the Nature of the Quadratic Form To classify the nature of the quadratic form: 1. **If the determinant is positive and both $$a$$ and $$b$$ are positive**, the form is **positive definite**. 2. **If the determinant is positive and both $$a$$ and $$b$$ are negative**, the form is **negative definite**. 3. **If the determinant is negative**, the form is **indefinite**. 4. **If the determinant is zero**, the form is **degenerate**. Here: - The determinant is $$5$$, which is positive. - Both $$a = 2$$ and $$b = 3$$ are positive. Thus, the quadratic form $$2x^2 + 3y^2 + 2xy + 22$$ is **positive definite**. Would you like more details, or do you have any questions? ### Related Questions 1. What is the significance of the discriminant in determining the nature of a quadratic form? 2. How would the nature of the quadratic form change if one of the coefficients was negative? 3. How do we handle quadratic forms with three or more variables? 4. What is the geometric interpretation of a positive definite quadratic form? 5. How do eigenvalues relate to the nature of a quadratic form? #### Tip For any quadratic form, checking the definiteness can provide insights into its graph shape (e.g., paraboloids or hyperboloids in 3D).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Quadratic Forms
Matrix Theory
Formulas
Quadratic form matrix representation: ax^2 + by^2 + 2hxy
Determinant formula for a 2x2 matrix: det(A) = ab - h^2
Theorems
Definiteness of a Quadratic Form
Suitable Grade Level
Undergraduate
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