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## GR0877 Problem 86

PROBLEM STATEMENT: This problem is still being typed.

SOLUTION: (B) The eigenvalues of the matrix $A$ are the solutions $\lambda$ to the equation $\det(A - \lambda I) = 0$, where $I$ is identity matrix. Thus, $A -\lambda I = \begin{pmatrix} 2 - \lambda & i\\ -i & 2- \lambda \end{pmatrix}$ and $\det(A - \lambda I) = (2-\lambda)^2 + i^2 \equiv 0$. From this, one has $\lambda_1 = 1$, $\lambda_2 = 3$.

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The trace of any matrix is invariant, so $Tr(A) = \Sigma \lambda_i$. This rules out A (and also D and E if you forgot the first property).
$\det(A)$ is simply the product of the eigenvalues. This leaves us with B