https://r2.erweima.ai/mathbot/image/1724382674287c9tca901.jpg

SI A⋅=|■(2&0.5&Y@X&-1&Z@-6&8&7)|ES UNAMATRIZ SIMETRICA,HALLAX+Y

Given the matrix \( A = \begin{pmatrix} 2 & 0.5 & y \\ x & -1 & z \\ -6 & 8 & 7 \end{pmatrix} \), we need to find \( x + y \) assuming that \( A \) is a symmetric matrix.

### Understanding Symmetric Matrices
A matrix is symmetric if it is equal to its transpose. In other words, \( A \) is symmetric if \( A = A^T \). For matrix \( A \), this means that:

\[
a_{ij} = a_{ji} \quad \text{for all } i, j
\]

### Applying the Symmetry Condition
The symmetry condition applied to each relevant pair of elements in matrix \( A \) gives us the following equations:

1. \( a_{12} = a_{21} \): \( 0.5 = x \)
2. \( a_{13} = a_{31} \): \( y = -6 \)
3. \( a_{23} = a_{32} \): \( z = 8 \)

### Solving for \( x + y \)
From the first condition, \( x = 0.5 \).

From the second condition, \( y = -6 \).

Therefore,

\[
x + y = 0.5 + (-6) = -5.5
\]

### Final Answer
The value of \( x + y \) is \( \boxed{-5.5} \).

Would you like more details on any step? Here are 5 related questions:

1. What is a symmetric matrix, and why is it important in linear algebra?
2. Can a non-square matrix be symmetric?
3. How do you determine if a matrix is symmetric by inspection?
4. What are some applications of symmetric matrices in real-world problems?
5. How would the problem change if the matrix were not symmetric?

**Tip:** When solving problems involving matrices, always check if any given conditions (like symmetry) can simplify your calculations.

Explore the concept of symmetric matrices through solving for x + y in a specific matrix A. Learn the conditions for matrix symmetry and how to apply them in solving linear algebra problems. Suitable for advanced mathematics students and enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation