In mathematics, particularly linear algebra, a **zero matrix** or **null matrix** is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of matrices, and is denoted by the symbol or followed by subscripts corresponding to the dimension of the matrix as the context sees fit.^{[1]}^{[2]}^{[3]} Some examples of zero matrices are

The set of matrices with entries in a ring K forms a ring . The zero matrix in is the matrix with all entries equal to , where is the additive identity in K.

The zero matrix is the additive identity in .^{[4]} That is, for all it satisfies the equation

There is exactly one zero matrix of any given dimension *m*×*n* (with entries from 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 is typically denoted by 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.

The zero matrix also represents the linear transformation which sends all the vectors to the zero vector.^{[5]} It 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.^{[6]}

In ordinary least squares regression, if there is a perfect fit to the data, the annihilator matrix is the zero matrix.