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Linear Algebra With Applications
Found in: Page 1
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 is a symmetric nxn matrix such that An=0, then A must be the zero matrix.

The given statement is TRUE.

See the step by step solution

Step by Step Solution

Step 1: Check whether the given statement is TRUE or FALSE

If λ is an eigenvalue of A, then λn is an eigenvalue of An.

The only eigenvalue of An is 0. Hence,λn=0 which implies that λ=0.

This shows that the only eigenvalue of A is 0.

Since A is symmetric, every singular value σ of A satisfies σ=λ for some eigenvalue λ of A. This implies that every singular value of A is 0.

If A=U VT is the singular value decomposition of A, then the statement implies that is the zero matrix. Hence,

A=VT =A0VT =0

Step 2: Final Answer

The given statement is TRUE.

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