Two eigenvalues of a 3 × 3 real matrix \(P\) are \((2 + \sqrt−1)\) and 3. The…
2016
Two eigenvalues of a 3 × 3 real matrix \(P\) are \((2 + \sqrt−1)\) and 3. The determinant of \(P\) is .
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Correct answer: 15
Key insight: for a real matrix, any nonreal eigenvalue appears with its complex conjugate.
The given nonreal eigenvalue is 2 + i, so its complex conjugate 2 - i is also an eigenvalue.
The third eigenvalue is 3 (real).
The determinant equals the product of the eigenvalues: (2 + i)(2 - i) × 3.
Compute (2 + i)(2 - i) = 4 - i^2 = 4 - (−1) = 5, then multiply by 3 to get 15.
Determinant = 15