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Consider the following matrix: \[ A = \begin{bmatrix} 2 & 3 \\ x & y \end{bmatrix} \] If the eigenvalues of $A$ are 4 and 8, then

1. x = 4, y = 10
2. x = 5, y = 8
3. x = -3, y = 9
4. x = -4, y = 10

Correct answer: x = -4, y = 10

Solution

The sum of eigenvalues is 4 + 8 = 12, so the trace gives 2 + y = 12, hence y = 10. The product is 4\cdot 8 = 32, so the determinant gives 2y - 3x = 32; substituting y = 10 gives 20 - 3x = 32, so x = -4.

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