To find the eigenvalues of the matrix \( M \), we need to solve the characteristic equation:
\[ \text{det}(M - \lambda I) = 0 \]
where \( \lambda \) represents the eigenvalues and \( I \) is the identity matrix.
The given matrix \( M \) is:
\[ M = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix} \]
The characteristic equation becomes:
\[ \text{det}\left( \begin{bmatrix} 2-\lambda & -1 \\ 3 & 1-\lambda \end{bmatrix} \right) = (2-\lambda)(1-\lambda) - (-1)(3) = 0 \]
Expanding and solving this equation gives us:
\[ (2-\lambda)(1-\lambda) + 3 = 0 \]
\[ (2-\lambda)(1-\lambda) = -3 \]
\[ 2 - 2\lambda - \lambda + \lambda^2 = -3 \]
\[ \lambda^2 - 3\lambda + 5 = 0 \]
Using the quadratic formula, we find the solutions for \( \lambda \):
\[ \lambda = \frac{-(-4) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} \]
\[ \lambda = \frac{4 \pm \sqrt{9 - 20}}{2} \]
\[ \lambda = \frac{4 \pm \sqrt{-11}}{2} \]
\[ \lambda = \frac{4 \pm \sqrt{11}\;\;i}{2} \]
So, the eigenvalues of matrix \( M \) are complex conjugate pairs, where the real part is zero and the imaginary part is non-zero.
Therefore, the correct statement is:
B. The eigenvalues of M are complex conjugate pairs.
