Two eigenvalues of a $3 \times 3$ real matrix $P$ are $(2 + \sqrt{-1})$ and $3$. The determinant of $P$ is ________ .

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

Given:

• One eigenvalue is $\lambda_1 = 2 + i$ (where $i = \sqrt{-1}$).

• Since the matrix has real entries, its complex conjugate $\lambda_2 = 2 - i$ must also be an eigenvalue.

• The third eigenvalue is given as $\lambda_3 = 3$.

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

Thus, the determinant of $P$ is 15.