What is the nullity of a zero matrix?

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In summary, the nullity of an nxn zero matrix is n, as the dimension of its nullspace is n. This is because every vector satisfies the equation Ox=0, making the nullspace the entire \mathbb{R}^n. It is important to note that the only matrices with nullity 0 are invertible matrices, which the zero matrix is not.
  • #1
SprucerMoose
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Gday,

I was wondering if someone could tell what the nullity of an nxn zero matrix is? I can't decide if its zero or n. Could someone knowledgeable please enlighten me?Thanks
 
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  • #2
Hi SprucerMoose :smile:

Well, the nullity of a matrix is defined as the dimension of it's nullspace (or kernel). So let O be our zero matrix, then the nullspace is

[tex]\{x\in \mathbb{R}^n~\vert~Ox=0\}[/tex]

Clearly, every vector satisfies Ox=0. Thus the nullspace is entire [tex]\mathbb{R}^n[/tex]. The dimension of [tex]\mathbb{R}^n[/tex] is n. Hence, the nullity of the zero matrix is n.

Please note, that the matrices with nullity 0 are exactly the invertible matrices (in finite-dimensional spaces of course). And, as you might know, the zero matrix is far from being invertible!
 
  • #3
Thanks very much for the speedy and clear response.
 

1. What is the nullity of the zero matrix?

The nullity of the zero matrix is defined as the dimension of the null space, or the set of all vectors that when multiplied by the zero matrix result in the zero vector.

2. How is the nullity of the zero matrix related to its rank?

The nullity of the zero matrix is equal to the number of columns in the matrix, while its rank is equal to the number of linearly independent rows. Since the zero matrix has no linearly independent rows, its rank is always zero.

3. What is the significance of the nullity of the zero matrix?

The nullity of the zero matrix is an important concept in linear algebra as it helps us understand the properties of matrices and their transformations. It also plays a crucial role in solving systems of linear equations and determining the existence of solutions.

4. Can the nullity of the zero matrix ever be greater than zero?

No, the nullity of the zero matrix is always equal to zero. This is because the zero matrix has no linearly independent rows, and thus the null space is empty.

5. How can the nullity of the zero matrix be calculated?

The nullity of the zero matrix can be calculated by finding the dimension of the null space using Gaussian elimination or by using the rank-nullity theorem, which states that the nullity of a matrix is equal to the difference between the number of columns and the rank of the matrix.

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