Zero matrix

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In matrix algebra, the zero matrix is a matrix which has all entries equal to zero. The zero matrix acts as an absorbing element for matrix multiplication and as the additive identity for matrix addition. It represents the zero linear map.

The zero matrix may be denoted \scriptstyle 0_{m,n} for an m×n matrix, so (for example)

0_{3,2} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix}

It is evident that for any m×n matrix A,

\scriptstyle A + 0_{m,n} = A .\,

As with the identity matrix, the subscript may be omitted if the context admits only one zero matrix. In this example, any other zero matrix could not be added to A, so the subscript is redundant and we could equally have written

A + 0 = A .\,

It is also clear that the product of any matrix with a zero matrix is another zero matrix of the appropriate dimensions:

0_{m,n} A_{n,p} = 0_{m,p} \mbox{ and } A_{k,m} 0_{m,n} = 0_{k,n} .\,
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