Math Problem Statement
Solution
To determine if a matrix is symmetric, we need to understand the defining property of symmetric matrices. A matrix is symmetric if and only if , meaning the matrix is equal to its transpose.
Let's analyze the options provided:
- : This condition describes an orthogonal matrix, where the product of the matrix and its transpose yields the identity matrix, .
- : This is the correct condition for a symmetric matrix.
- : This implies the product yields the negative identity matrix, which does not define a symmetric matrix.
- : This describes a skew-symmetric matrix, where the matrix is equal to the negative of its transpose.
Conclusion:
The correct answer is:
- Option 2:
Would you like a more detailed explanation on symmetric matrices?
Here are five related questions to expand on this topic:
- What are the properties of symmetric matrices in terms of eigenvalues and eigenvectors?
- How does symmetry in a matrix affect its determinant?
- What is the relationship between symmetric matrices and positive-definite matrices?
- Can a non-square matrix be symmetric? Why or why not?
- 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
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