If the sum and product of eigenvalues of a 2 × 2 real matrix $\left[\begin{array}{cc}3& p\\ p& q\end{array}\right]$ are 4 and -1 respectively, then |p| is _______ (in integer).

The correct answer is 2.

Let the given 2 × 2 real matrix be represented by:
$A=\left[\begin{array}{cc}3& p\\ p& q\end{array}\right]$

We are given the following two properties of the eigenvalues of matrix $A$:
1. The sum of the eigenvalues is 4.
2. The product of the eigenvalues is -1.

Let us analyze these properties using standard linear algebra theorems:

Step 1: Relate the sum of eigenvalues to the trace of the matrix
The sum of the eigenvalues of any square matrix is equal to the trace of the matrix (the sum of the main diagonal elements).
For matrix $A$, the trace is:
$\text{Trace}\left(A\right)=3+q$

Setting this equal to the sum of the eigenvalues, we have:
$3+q=4$
Solving for $q$:
$q=4-3=1$

Step 2: Relate the product of eigenvalues to the determinant of the matrix
The product of the eigenvalues of any square matrix is equal to the determinant of the matrix.
For matrix $A$, the determinant is:
$det\left(A\right)=(3\times q)-(p\times p)=3q-p^2$

Setting this equal to the product of the eigenvalues, we get:
$3q-p^2=-1$

Step 3: Solve for |p|
Substitute the value of $q=1$ into the determinant equation:
$3\left(1\right)-p^2=-1$
Simplify the equation:
$3-p^2=-1$
Rearranging to solve for $p^2$:
$p^2=3+1=4$
Taking the square root on both sides:
$p=\pm 2$

Since the question asks for the absolute value of $p$ (which is |p|):
$\left|p\right|=2$