Short Answer

If A is a symmetric nxn matrix such that $A^{n} = 0$ , then A must be the zero matrix.

The given statement is TRUE.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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

If $λ$ is an eigenvalue of A, then $λ^{n}$ is an eigenvalue of $A^{n}$ .

The only eigenvalue of $A^{n}$ is 0. Hence, $λ^{n} = 0$ 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 $A = U \sum_{} V^{T}$ is the singular value decomposition of A, then the statement implies that $\sum_{}$ is the zero matrix. Hence,

$A = \sum V^{T} = A (0) V^{T} = 0$

Step 2: Final Answer

The given statement is TRUE.

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Linear Algebra With Applications

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Short Answer

If A is a symmetric nxn matrix such that $A^{n} = 0$ , then A must be the zero matrix.

Step by Step Solution

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

Step 2: Final Answer

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Linear Algebra With Applications

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Short Answer

If A is a symmetric nxn matrix such that An=0, then A must be the zero matrix.

Step by Step Solution

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

Step 2: Final Answer

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If A is a symmetric nxn matrix such that $A^{n} = 0$ , then A must be the zero matrix.