Given Matrix:
The given matrix is:
1 3 -1

Singular Matrix:
A matrix is said to be singular if its determinant is zero.

Determinant Calculation:
To determine if the given matrix is singular or not, we need to calculate its determinant.

|1 3|
|-1 x|

The determinant of a 2x2 matrix can be calculated as:
Determinant = (a*d) - (b*c)

In this case, the determinant is:
Determinant = (1 * x) - (3 * -1)
Determinant = x + 3

Condition for Singularity:
As per the given problem, the matrix is singular. Therefore, the determinant should be equal to zero.

So, x + 3 = 0
x = -3

Matrix with x = -3:
The given matrix with x = -3 is:
1 3 -1
-1 -3 2
4 3 2
-3

Explanation:
To find the value of x in the matrix 1 3 -1 x 0 2 4 3 2 x, we need to determine if the matrix 1 3 -1 is singular or not.

To check for singularity, we calculate the determinant of the 2x2 sub-matrix formed by removing the last row and last column. In this case, the determinant is calculated as x + 3.

Since the matrix is singular, the determinant should be equal to zero. Therefore, we equate x + 3 to zero and solve for x. The value of x is determined to be -3.

Substituting x = -3 into the given matrix, we obtain the matrix:
1 3 -1
-1 -3 2
4 3 2
-3

This is the updated matrix with x = -3.