**Matrix addition** is only defined for matrices with the same dimensions. This is because each corresponding entry is added. If $A=(a_{ij})\in F^{m\times n}$ and $B=(b_{ij})\in F^{m\times n}$, then $A+B=C=(c_{ij})\in F^{m\times n}$ is defined by $c_{ij}=a_{ij}+b_{ij}$ The [[zero matrix]] of appropriate dimensions functions as the [[additive identity]].