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3: A matrix Xx € Mnxn is called symmetric if XT = X, and skew-symmetric if XT = -X. a) Show that the diagonal entries of a skew-symmetric matrix are 0. b) Let A € Mnxn be any square matrix. Define X = (A+ A") a + and Y = (A – A"). Show that X is symmetric and Y is skew-symmetric. c) Show that any nx n square matrix can be written as a sum of a symmetric and a skew symmetric matrix.

3: A matrix Xx € Mnxn is called symmetric if XT = X, and skew-symmetric if XT = -X. a) Show that the diagonal entries of a skew-symmetric matrix are 0. b) Let A € Mnxn be any square matrix. Define X = (A+ A") a + and Y = (A – A"). Show that X is symmetric and Y is skew-symmetric. c) Show that any nx n square matrix can be written as a sum of a symmetric and a skew symmetric matrix.

Algebra and Trigonometry (6th Edition)
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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**Matrix Symmetry Question** A matrix \( X \in M_{n \times n} \) is called symmetric if \( X^T = X \), and skew-symmetric if \( X^T = -X \). a) **Diagonal Entries of Skew-Symmetric Matrix** Show that the diagonal entries of a skew-symmetric matrix are 0. b) **Symmetric and Skew-Symmetric Matrices from a Square Matrix** Let \( A \in M_{n \times n} \) be any square matrix. Define \( X = \frac{1}{2}(A + A^T) \) and \( Y = \frac{1}{2}(A - A^T) \). Show that \( X \) is symmetric and \( Y \) is skew-symmetric. c) **Decomposition of a Square Matrix** Show that any \( n \times n \) square matrix can be written as a sum of a symmetric and a skew-symmetric matrix.
Transcribed Image Text:**Matrix Symmetry Question** A matrix \( X \in M_{n \times n} \) is called symmetric if \( X^T = X \), and skew-symmetric if \( X^T = -X \). a) **Diagonal Entries of Skew-Symmetric Matrix** Show that the diagonal entries of a skew-symmetric matrix are 0. b) **Symmetric and Skew-Symmetric Matrices from a Square Matrix** Let \( A \in M_{n \times n} \) be any square matrix. Define \( X = \frac{1}{2}(A + A^T) \) and \( Y = \frac{1}{2}(A - A^T) \). Show that \( X \) is symmetric and \( Y \) is skew-symmetric. c) **Decomposition of a Square Matrix** Show that any \( n \times n \) square matrix can be written as a sum of a symmetric and a skew-symmetric matrix.
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