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Given \(A=\begin{bmatrix}a&b&c\\ b&c&a\\ c&a&b\end{bmatrix}\) and \(AA^T=I\), the values of \(a\), \(b\), and \(c\) must satisfy which of the following equations?

1. \(x^3+abc=0\)
2. \(x^3+x^2-abc=0\)
3. \(x^3-2x^2+abc=0\)
4. \(x^3\pm x^2+abc=0\)

Correct answer: \(x^3-2x^2+abc=0\)

Solution

Since \(AA^T=I\), the matrix is orthogonal, so its rows are orthonormal and its determinant has magnitude 1. For this cyclic matrix, the algebraic relation among the entries leads to the cubic equation \(x^3-2x^2+abc=0\).

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