> [!Definition] Definition (Zero matrix) > The **zero matrix**, denoted $0_{mn}$ , is the $m\times n$ whose entries are all $0$. >> > # Properties The zero matrix is the [[Identity element of a binary operation|identity element]] of [[Matrix Scalar Multiplication|matrix addition]], i.e. $A+0_{mn}=0_{mn}+A=A.$ The [[Matrix Product|product]] of any matrix and a zero is a zero matrix. I.e. let $A\in \text{M}_{mn}$ and $l,p\in\mathbb{Z}^{+}$. Then $A 0_{np}=0_{mp}$ and $0_{lp}A=0_{\ln}$. >*Proof*. To find an entry of the product $A0_{np}$ we [[Dot Product in Real n-Space|dot]] a row of $A$ with a zero column of $0_{np}$, which will always give zero.