Zero matrix

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 formula_1 matrices, and is denoted by the symbol formula_2 or formula_3—followed by subscripts corresponding to the dimension of the matrix as the context sees fit. Some examples of zero matrices are
Properties.
The set of formula_1 matrices with entries in a ring K forms a ring formula_6. The zero matrix formula_7 in formula_8 is the matrix with all entries equal to formula_9, where formula_10 is the additive identity in K. 
The zero matrix is the additive identity in formula_8. That is, for all formula_13 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. 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.
Occurrences.
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.
In ordinary least squares regression, if there is a perfect fit to the data, the annihilator matrix is the zero matrix.