### Quadratic Forms in Linear AlgebraIn this educational activity, you will learn how to compute the quadratic form $ \mathbf{x}^T A \mathbf{x} $ for a given matrix $ A $ and a set of vectors $ \mathbf{x} $. The quadratic form is an important concept frequently encountered in various fields such as statistics, physics, and engineering.#### Given Matrix:\[ A = \begin{bmatrix}5 & 6 & 0 \\6 & 3 & 1 \\0 & 1 & 4\end{bmatrix} \]#### Vectors:1. $ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} $2. $ \mathbf{x} = \begin{bmatrix} 6 \\ 5 \\ 7 \end{bmatrix} $3. $ \mathbf{x} = \begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{-1}{\sqrt{3}} \end{bmatrix} $### Steps to Follow:1. **Compute** the quadratic form $ \mathbf{x}^T A \mathbf{x} $ for each vector.2. **Simplify** your answer.#### Example Computations:##### a. For $ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} $,\[ \mathbf{x}^T A \mathbf{x} = \_ \]##### b. For $ \mathbf{x} = \begin{bmatrix} 6 \\ 5 \\ 7 \end{bmatrix} $,\[ \mathbf{x}^T A \mathbf{x} = \_ \]*(Simplify your answer.)*##### c. For $ \mathbf{x} = \begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{-1}{\sqrt{3}} \end{bmatrix} $,\[ \mathbf{x}^T A \mathbf{x} = \_ \]*(Simplify your answer.)***Note**: Ensure each step is clearly shown to help in understanding the computations involved. The process for computing

Transpose of each column vector is a row vector. We do simple matrix multiplication to compute the…

Answered: 5 6 0 Compute the quadratic form x'Ax…

5 6 0 Compute the quadratic form x'Ax for A = 6 3 1 and each of the following vectors. 0 14 X1 1/ 3 6 a. X= X2 b. x= 5 C. X = 1/ 3 X3 -1/3

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

### Quadratic Forms in Linear Algebra

In this educational activity, you will learn how to compute the quadratic form $ \mathbf{x}^T A \mathbf{x} $ for a given matrix $ A $ and a set of vectors $ \mathbf{x} $. The quadratic form is an important concept frequently encountered in various fields such as statistics, physics, and engineering.

#### Given Matrix:
\[ A = \begin{bmatrix}
5 & 6 & 0 \\
6 & 3 & 1 \\
0 & 1 & 4
\end{bmatrix} \]

#### Vectors:
1. $ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} $
2. $ \mathbf{x} = \begin{bmatrix} 6 \\ 5 \\ 7 \end{bmatrix} $
3. $ \mathbf{x} = \begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{-1}{\sqrt{3}} \end{bmatrix} $

### Steps to Follow:
1. **Compute** the quadratic form $ \mathbf{x}^T A \mathbf{x} $ for each vector.
2. **Simplify** your answer.

#### Example Computations:
##### a. For $ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} $,
\[ \mathbf{x}^T A \mathbf{x} = \_ \]

##### b. For $ \mathbf{x} = \begin{bmatrix} 6 \\ 5 \\ 7 \end{bmatrix} $,
\[ \mathbf{x}^T A \mathbf{x} = \_ \]
*(Simplify your answer.)*

##### c. For $ \mathbf{x} = \begin{bmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \\ \frac{-1}{\sqrt{3}} \end{bmatrix} $,
\[ \mathbf{x}^T A \mathbf{x} = \_ \]
*(Simplify your answer.)*

**Note**: Ensure each step is clearly shown to help in understanding the computations involved. The process for computing

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