Question-6

Consider the matrix \(\mathbf{M} = \begin{bmatrix}2 & -1\\3 & 1\end{bmatrix}\). Which one of the following statements is true?

• The eigenvalues of \(\mathbf{M}\) are non-negative and real.
• The eigenvalues of \(\mathbf{M}\) are complex conjugate pairs.
• One eigenvalue of \(\mathbf{M}\) is positive and real, and another eigenvalue of \(\mathbf{M}\) is zero.
• One eigenvalue of \(\mathbf{M}\) is non-negative and real, and another eigenvalue of \(\mathbf{M}\) is negative and real

Answer

• The eigenvalues of \(\mathbf{M}\) are non-negative and real.
• The eigenvalues of \(\mathbf{M}\) are complex conjugate pairs.
• One eigenvalue of \(\mathbf{M}\) is positive and real, and another eigenvalue of \(\mathbf{M}\) is zero.
• One eigenvalue of \(\mathbf{M}\) is non-negative and real, and another eigenvalue of \(\mathbf{M}\) is negative and real

Solution

We can compute the eigenvalues with the help of the characteristic polynomial: \[ \begin{aligned} \begin{vmatrix} 2 - \lambda & -1\\ 3 & 1 - \lambda \end{vmatrix} &= 0\\\\ (2 - \lambda)(1 - \lambda) + 3 &= 0\\\\ \lambda^2 -3 \lambda + 5 &= 0\\\\ \end{aligned} \] We see that the discriminant of the quadratic equation is negative. Hence, the eigenvalues are complex conjugate pairs.