There is exactly one zero matrix of any given size m×n having entries in a given ring, so when the context is clear one often refers to the zero matrix. In general the zero element of a ring is unique and typically denoted as 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.
The zero matrix is idempotent, meaning that when it is multiplied by itself the result is itself.
The zero matrix is the only matrix whose rank is 0.
The mortal matrix problem is the problem of determining, given a finite set of n × n matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix. This is known to be undecidable for a set of six or more 3 × 3 matrices, or a set of two 15 × 15 matrices.
- Lang, Serge (1987), Linear Algebra, Undergraduate Texts in Mathematics, Springer, p. 25, ISBN 9780387964126,
We have a zero matrix in which aij = 0 for all i, j. ... We shall write it O.
- Warner, Seth (1990), Modern Algebra, Courier Dover Publications, p. 291, ISBN 9780486663418,
The neutral element for addition is called the zero matrix, for all of its entries are zero.
- Bronson, Richard; Costa, Gabriel B. (2007), Linear Algebra: An Introduction, Academic Press, p. 377, ISBN 9780120887842,
The zero matrix represents the zero transformation 0, having the property 0(v) = 0 for every vector v ∈ V.
- Cassaigne, Julien; Halava, Vesa; Harju, Tero; Nicolas, Francois (2014). "Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More". arXiv:1404.0644 [cs.DM].