### 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 \\

Answered: Image /qna-images/answer/a0e2b5fb-4171-49c7-b30a-56d828691bf0.jpg

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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Question

### 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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