**If AX=0 has a non-zero solution then A is**

**A. singular**

**B. non-singular**

**C zero**

**D none**

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.

Explore why a matrix is considered singular when the equation AX = 0 has a non-zero solution. This detailed explanation covers matrix theory, determinant properties, and the implications of singularity in linear algebra. Suitable for undergraduate students and anyone interested in matrix theory.

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