The correct answer is A. singular.

Here's why:

• The equation $AX = 0$ represents a homogeneous system of linear equations. If this system has a non-zero solution $X$, then the matrix $A$ must be singular.

• A matrix $A$ is singular if its determinant is zero, which means it does not have an inverse. This also implies that the rows (or columns) of $A$ are linearly dependent, leading to the existence of a non-trivial solution to the equation $AX = 0$.

• If $A$ were non-singular (i.e., it has an inverse), the only solution to $AX = 0$ would be the trivial solution $X = 0$.

So, if $AX = 0$ has a non-zero (non-trivial) solution, matrix $A$ must be singular.

Would you like further details or have any other questions?

Here are 5 related questions to expand on this concept:

1. How can you determine if a matrix is singular or non-singular?
2. What is the relationship between the determinant of a matrix and the existence of non-trivial solutions to $AX = 0$?
3. What does it mean for a system of linear equations to be homogeneous?
4. How does the rank of a matrix relate to its singularity?
5. Can a singular matrix have an inverse? Why or why not?

Tip: If you're given a matrix and asked whether it's singular, check if its determinant is zero. This is a quick and reliable method.