Let's consider the case of a zero matrix.

### Zero Matrix:
A zero matrix is a matrix where every entry is zero. For an \( m \times n \) zero matrix \( A \), let's analyze the rank and nullity:

- **Rank**: The rank of a matrix is the dimension of its column space (the space spanned by its columns). For the zero matrix, all columns are the zero vector, meaning they are linearly dependent and don't span any space. Thus, the rank of the zero matrix is \( 0 \).

- **Nullity**: The nullity is the dimension of the null space (the set of all vectors \( \mathbf{x} \) such that \( A\mathbf{x} = \mathbf{0} \)). For the zero matrix, \( A\mathbf{x} = \mathbf{0} \) for **any** vector \( \mathbf{x} \) in \( \mathbb{R}^n \). Therefore, the null space is the entire \( \mathbb{R}^n \), and the nullity is \( n \).

### Applying the Rank-Nullity Theorem:
For an \( m \times n \) zero matrix \( A \):
\[
\text{Rank}(A) + \text{Nullity}(A) = n
\]
Since \( \text{Rank}(A) = 0 \), it follows that \( \text{Nullity}(A) = n \).

### What Happens When \( n > m \)?
If \( n > m \), the matrix is "wide," meaning there are more columns than rows. For a zero matrix in this case:

- **Rank**: \( \text{Rank}(A) = 0 \) (as discussed).
- **Nullity**: \( \text{Nullity}(A) = n \).

So yes, in the special case of a zero matrix, it is indeed possible for an \( m \times n \) matrix \( A \) to have a nullity of \( n \) even when \( n > m \). This is because the zero matrix inherently satisfies \( A\mathbf{x} = \mathbf{0} \) for every vector \( \mathbf{x} \) in \( \mathbb{R}^n \), making the null space as large as possible.

Would you like more details or have any other questions?

Here are 5 related questions to consider:
1. How does the null space of a non-zero matrix compare to that of a zero matrix?
2. What is the impact of a zero matrix on solving systems of linear equations?
3. How does the dimension of the null space affect the number of solutions to the system \( A\mathbf{x} = \mathbf{b} \) where \( A \) is a zero matrix?
4. Can a zero matrix be used to determine the linear dependence of a set of vectors?
5. What are the implications of having a matrix with a rank of zero?

**Tip:** The zero matrix is a unique case in linear algebra where the concepts of rank and nullity

Explore the concept of a zero matrix in linear algebra, its properties, and implications. Learn about the Rank-Nullity Theorem and how it applies to matrices. Suitable for college-level students and anyone interested in advanced linear algebra concepts.

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