Math Problem Statement

A matrix A is symmetric if and only if...

Solution

To determine if a matrix AA is symmetric, we need to understand the defining property of symmetric matrices. A matrix AA is symmetric if and only if A=ATA = A^T, meaning the matrix is equal to its transpose.

Let's analyze the options provided:

  1. AAT=ATA=IA A^T = A^T A = I: This condition describes an orthogonal matrix, where the product of the matrix and its transpose yields the identity matrix, II.
  2. A=ATA = A^T: This is the correct condition for a symmetric matrix.
  3. AAT=ATA=IA A^T = A^T A = -I: This implies the product yields the negative identity matrix, which does not define a symmetric matrix.
  4. A=ATA = -A^T: This describes a skew-symmetric matrix, where the matrix is equal to the negative of its transpose.

Conclusion:

The correct answer is:

  • Option 2: A=ATA = A^T

Would you like a more detailed explanation on symmetric matrices?


Here are five related questions to expand on this topic:

  1. What are the properties of symmetric matrices in terms of eigenvalues and eigenvectors?
  2. How does symmetry in a matrix affect its determinant?
  3. What is the relationship between symmetric matrices and positive-definite matrices?
  4. Can a non-square matrix be symmetric? Why or why not?
  5. How do symmetric matrices relate to quadratic forms in linear algebra?

Tip: A symmetric matrix always has real eigenvalues, which is a key property in many applications like optimization and stability analysis.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Theory
Symmetric Matrices

Formulas

A = A^T

Theorems

Symmetric Matrix Property

Suitable Grade Level

Undergraduate Level