What is the nullity of a zero matrix?

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SUMMARY

The nullity of an nxn zero matrix is definitively n. The nullity is defined as the dimension of its nullspace, which consists of all vectors x in ℝ^n that satisfy the equation Ox=0. Since every vector in ℝ^n satisfies this equation for the zero matrix O, the nullspace is the entirety of ℝ^n, resulting in a nullity of n. In contrast, invertible matrices have a nullity of 0, highlighting the non-invertibility of the zero matrix.

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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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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

\{x\in \mathbb{R}^n~\vert~Ox=0\}

Clearly, every vector satisfies Ox=0. Thus the nullspace is entire \mathbb{R}^n. The dimension of \mathbb{R}^n 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!
 
Thanks very much for the speedy and clear response.
 

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