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Chapter 5: Q41E (page 264)

TRUE OR FALSE

41. Determine whether the statement 鈥淎ny square matrix can be written as the sum of the symmetric and a skew-symmetric matrix.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of a symmetric matrix

鈥淢atrix A is symmetric if AT=A.

02

Explanation of the solution

Consider the statement as follows.

鈥淎ny square matrix can be written as the sum of a symmetric and skew-symmetric matrix.

Matrix A is symmetric if AT=A.

Matrix A is skew-symmetric if .AT=鈭�A

Then (A+AT)is a symmetric matrix and (A鈭�AT)is a skew-symmetric matrix.

Then A can be written as follows.

A=12(A+AT)+12(A-AT)=12A+12AT+12A鈭�12AT=12A+12A=A

This, implies that A is the sum of symmetric and skew-symmetric matrix.

Hence, the given statement 鈥淎ny square matrix can be written as the sum of a symmetric and skew-symmetric matrix is true.

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