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

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

(GATE 2016) 

Answer: 15 

Explanation: 

Here, $P$ is a $3 \times 3$ real matrix. 

The given eigenvalues are $2 + i$ and $3$. 

Since $P$ is a real matrix, the third eigenvalue must be $2 - i$. 

We know that the determinant of a real matrix is equal to the product of its eigenvalues. 

Therefore, the determinant of $P$ = product of eigenvalues = $(2 + i) \times 3 \times (2 - i)$ = $15$.