GATE CS 2016 Set 1 Q05 — Eigenvalues and Eigenvectors, LU Decomposition

For any real matrix, complex eigenvalues must always occur in conjugate pairs.

We are given two eigenvalues of the $3 \times 3$ real matrix $P$:

• $\lambda_1 = 2 + \sqrt{-1} = 2 + i$
• $\lambda_2 = 3$

Since $P$ is a real matrix, the complex conjugate of $\lambda_1$ must also be an eigenvalue of $P$:

The determinant of a matrix is equal to the product of all its eigenvalues:

Thus, the determinant of $P$ is 15.