Find the remaining eigenvalue for matrix \( A=\left[\begin{array}{rr}3 & -5 \\ 8 & -9\end{array}\right] \) \( \lambda=-3+2 i \).

To find the remaining eigenvalue of the matrix \( A=\left[\begin{array}{rr}3 & -5 \\ 8 & -9\end{array}\right] \) given one eigenvalue \( \lambda=-3+2 i \), we can use the property that for a matrix with real entries, the eigenvalues occur in complex conjugate pairs. Thus, the other eigenvalue must be the complex conjugate of the given eigenvalue. Therefore, if \( \lambda_1 = -3 + 2i \) is one eigenvalue, then the remaining eigenvalue \( \lambda_2 \) will be: \[ \lambda_2 = -3 - 2i \] So, the remaining eigenvalue is \( -3 - 2i \). And there you have it; eigenvalues can be quite the duo!