What is the rank and nullity of the zero matrix?

Question

What is the rank and nullity of the $2 \times 2$ matrix:

\begin{bmatrix} 0 & 0\\ 0 & 0\\ \end{bmatrix}

Answer 1

The rank of a matrix is the dimension of the column space, the linear subspace of the codomain spanned by the columns.

For a matrix whose only entries are zero, the column space would be spanned only by zero vectors. Any linear combination of zero vectors is again a zero vector.

The space containing only the zero vector and no others is considered to be zero-dimensional. The rank is then zero.

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The nullity is the dimension of the nullspace, the subspace of the domain consisting of all vectors from the domain who when the matrix is applied to it result in the zero vector. It is clear that for $Z$ a zero matrix and any vector $v$ in the domain that $Zv=\vec{0}$ results in the zero vector and so the nullspace is the entire domain.

As such, the nullity of any matrix containing all zeroes would be the number of columns of the matrix, i.e. the dimension of the domain.

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TLDR: The nullity of $\begin{bmatrix}0&0\\0&0\end{bmatrix}$ is $2$ while the rank is $0$.