A matrix is singular iff it has_______.
A) two eigen value. 
B)one eigen value. 
C) none of above
D) zero eigen value.

A matrix is singular if and only if it has a **zero eigenvalue**. 

### Explanation:
- A matrix is considered singular if its determinant is zero.
- If a matrix has a zero eigenvalue, it means there is a non-trivial solution to the equation \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \) where \( \lambda = 0 \). This implies that the matrix \( \mathbf{A} \) does not have full rank, leading to a determinant of zero.
- Conversely, if a matrix has a non-zero determinant, it means none of its eigenvalues are zero, and the matrix is non-singular.

### Correct Answer:
**D) Zero eigenvalue.**

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Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How can you determine the eigenvalues of a given matrix?
2. What is the relationship between the determinant and the eigenvalues of a matrix?
3. How does the rank of a matrix relate to its singularity?
4. Can a matrix have more than one zero eigenvalue, and what does that imply?
5. What is the geometric interpretation of an eigenvalue being zero?

**Tip:** Remember that a matrix being singular directly ties to its determinant, and checking for zero eigenvalues is a key method in linear algebra to assess singularity.

Explore the concept of matrix singularity and understand when a matrix is considered singular. Learn about eigenvalues and their relationship with matrix determinants. Suitable for advanced high school students and above.

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