Short Answer

47. If $A = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$ , then there exist exactly four orthogonal $2 \times 2$ matrices S such that $S^{- 1}$ AS is diagonal.

The given statement is FALSE.

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Step by Step Solution

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TABLE OF CONTENTS

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

Since is symmetric, it has an orthogonal basis consisting of eigenvectors. Let ${\dot{v}}_{1}, {\dot{v}}_{2}$ be unit orthogonal eigenvectors of A. Since S is orthogonal diagonal zing matrix of A,S must map $\{e_{1}, e_{2}\}$ to orthogonal eigenvectors.

Thus, we have either ${Se}_{1} = \pm {\dot{v}}_{1}$ and ${Se}_{2} = \pm {\dot{v}}_{2}$ OR ${Se}_{1} = \pm {\dot{v}}_{2}$ and ${Se}_{2} = \pm {\dot{v}}_{1}$ .

This gives a total of 8 possibilities for S.

Step 2: Final Answer

Therefore, the given statement is FALSE.

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Linear Algebra With Applications

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Short Answer

47. If $A = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$ , then there exist exactly four orthogonal $2 \times 2$ matrices S such that $S^{- 1}$ AS is diagonal.

Step by Step Solution

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

Step 2: Final Answer

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47. If A=[1223], then there exist exactly four orthogonal 2×2matrices S such that S-1AS is diagonal.

Step by Step Solution

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

Step 2: Final Answer

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47. If $A = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$ , then there exist exactly four orthogonal $2 \times 2$ matrices S such that $S^{- 1}$ AS is diagonal.