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Expert-verified Found in: Page 1 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # If A is a symmetric nxn matrix such that ${{\mathbf{A}}}^{{\mathbf{n}}}{\mathbf{=}}{\mathbf{0}}$, then A must be the zero matrix.

The given statement is TRUE.

See the step by step solution

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

If $\lambda$ is an eigenvalue of A, then ${\mathrm{\lambda }}^{\mathrm{n}}$ is an eigenvalue of ${\mathrm{A}}^{\mathrm{n}}$.

The only eigenvalue of ${\mathrm{A}}^{\mathrm{n}}$ is 0. Hence,${\lambda }^{n}=0$ which implies that $\lambda =0$.

This shows that the only eigenvalue of A is 0.

Since A is symmetric, every singular value $\sigma$ of A satisfies $\mathrm{\sigma }=\left|\mathrm{\lambda }\right|$ for some eigenvalue $\lambda$ of A. This implies that every singular value of A is 0.

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

The given statement is TRUE. ### Want to see more solutions like these? 