### Quadratic Forms in Linear Algebra**Problem Statement:**Find the matrix \(A\) of the quadratic form associated with the equation:\[ 10x^2 + 10y^2 + 10z^2 + 10xy + 10xz + 10yz - 1 = 0 \]\[\mathbf{A} = \begin{bmatrix}& & \\& & \\& &\end{bmatrix}\]Find the equation of the rotated quadratic surface in which the \(xy\)-, \(xz\)-, and \(yz\)-terms have been eliminated. (Use \(xp\), \(yp\), and \(zp\) as the new coordinates.) \[\boxed{}\]**Solution Steps:**1. **Formulating the Quadratic Form Matrix:** The given quadratic equation can be represented in a matrix form \( \mathbf{x}^\top \mathbf{A} \mathbf{x} \), where \( \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). To find the matrix \(\mathbf{A}\), we need to identify the coefficients of \(x^2\), \(y^2\), \(z^2\), \(xy\), \(xz\), and \(yz\). For our given equation: - The coefficient of \(x^2\) is 10. - The coefficient of \(y^2\) is 10. - The coefficient of \(z^2\) is 10. - The coefficient of \(xy\) is 10 (which means each \(A_{12}\) and \(A_{21}\) is \(\frac{10}{2}\)). - The coefficient of \(xz\) is 10 (which means each \(A_{13}\) and \(A_{31}\) is \(\frac{10}{2}\)). - The coefficient of \(yz\) is 10 (which means each \(A_{23}\) and \(A_{32}\) is \(\frac{10}{2}\)). This leads us to the matrix \( \mathbf{A} \): \[ \mathbf{A} = \begin{bmatrix} 10 & 5 & 5 \\ 5 & 10 & 5 \\

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Find the matrix A of the quadratic form associated with the equation. 10x2 + 10y? + 10z2 + 10xy + 10xz + 10yz - 1 = 0 A Find the equation of the rotated quadratic surface in which the xy-, xz-, and yz-terms have been eliminated. (Use xp, yp, and zp as the new coordinates.)

Find the matrix A of the quadratic form associated with the equation. 10x2 + 10y? + 10z2 + 10xy + 10xz + 10yz - 1 = 0 A Find the equation of the rotated quadratic surface in which the xy-, xz-, and yz-terms have been eliminated. (Use xp, yp, and zp as the new coordinates.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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