Question

If the sum and product of eigenvalues of a $2 \times 2$ real matrix $\begin{bmatrix} 3 & p \\ p & q \end{bmatrix}$ are 4 and $-1$ respectively, then $|p|$ is ____________ (in integer).

Hint:

Trace gives eigenvalue sum, determinant gives eigenvalue product—easy shortcut for $2 \times 2$ matrices.

Answer 1

Correct Answer: 2

For a $2\times2$ matrix, the sum of eigenvalues equals the trace:
\[ 3 + q = 4 \Rightarrow q = 1. \] The product of eigenvalues equals the determinant:
\[ \lambda_1 \lambda_2 = 3(1) - p^2 = -1. \] Thus,
\[ 3 - p^2 = -1 \Rightarrow p^2 = 4 \Rightarrow |p| = 2. \] Therefore, the answer is
\[ \boxed{2} \]