Discussion
The product of eigenvalues of the matrix P is
\[ P = \begin{bmatrix} 2 & 0 & 1 \\ 4 & -3 & 3 \\ 0 & 2 & -1 \end{bmatrix} \]
Answer: B
The product of the eigenvalues is 2.
The product of the eigenvalues of a square matrix equals its determinant. Expanding along the first row:
\[ \begin{aligned} |P| &= 2[(-3)(-1) - (3)(2)] - 0 + 1[(4)(2) - 0] \\ &= 2(3 - 6) + 8 = 2 \end{aligned} \]
-6 is the first term alone, \(2(3 - 6)\), without the contribution of the corner element 1; 6 and -2 come from sign slips in the cofactors. The sum of the eigenvalues would be the trace, 2 - 3 - 1 = -2, which is the other quantity the distractor -2 matches.
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