Final answer:
The matrix has distinct real eigenvalues as determined by the positive discriminant of its characteristic polynomial, indicating two distinct real roots.
Step-by-step explanation:
To determine whether the eigenvalues of a matrix are distinct real, repeated real, or complex, we need to find the roots of the characteristic polynomial of the matrix. The matrix in question is:
\[\begin{pmatrix}-5 & -2 \\ 8 & 3\end{pmatrix}\]
To find its eigenvalues, we calculate the determinant of the matrix subtracted by a scalar \(\lambda\) times the identity matrix, which gives us the characteristic equation:
\[\text{det}(A - \lambda I) = (\text{-5} - \lambda)(\text{3} - \lambda) - (\text{-2})(\text{8}) = 0\]
This simplifies to:
\[\lambda^2 + 2\lambda - 31 = 0\]
By solving this quadratic equation, we analyze the discriminant (\(b^2 - 4ac\)), which tells us the nature of the roots:
\[\text{Discriminant} = 2^2 - 4(1)(-31) = 4 + 124 = 128\]
Since the discriminant is positive, we have two distinct real roots. Thus, the eigenvalues of this matrix are distinct real numbers.