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Linear Algebra With Applications
Found in: Page 233
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

If A and B are arbitrary n×n matrices, which of the matrices in Exercise 21 through 26 must be symmetric?


The matrix ATBA is not symmetric.

See the step by step solution

Step by Step Solution

Step 1: Condition to be a symmetric.

A matrix is symmetric if and only if it is equal to its transpose.

All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal. A matrix is skew-symmetric if and only if it is the opposite of its transpose.

For example,

11-112 0-10 5

Thus, it gives


Therefore, if A and B are arbitrary n×n matrices then the matrices ATBA is not symmetric.

Hence, the matrix is not necessarily symmetric.

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