Let \(A,B\in M_{m\times n}(F)\) be two matrices over a given field. The matrix addition "$+$" is defined by
$$ C:=A+B=\pmatrix{ \alpha_{11} & \alpha_{12} & \ldots & \alpha_{1n} \cr \alpha_{21} & \alpha_{22} & \ldots & \alpha_{2n} \cr \vdots & \vdots & \ddots & \vdots \cr \alpha_{m1} & \alpha_{m2} & \ldots & \alpha_{mn} \cr }+\pmatrix{ \beta_{11} & \beta_{12} & \ldots & \beta_{1n} \cr \beta_{21} & \beta_{22} & \ldots & \beta_{2n} \cr \vdots & \vdots & \ddots & \vdots \cr \beta_{m1} & \beta_{m2} & \ldots & \beta_{mn} \cr }=\pmatrix{ \alpha_{11}+\beta_{11} & \alpha_{12}+\beta_{12} & \ldots & \alpha_{1n}+\beta_{1n} \cr \alpha_{21}+\beta_{21} & \alpha_{22}+\beta_{22} & \ldots & \alpha_{2n}+\beta_{2n} \cr \vdots & \vdots & \ddots & \vdots \cr \alpha_{m1}+\beta_{m1} & \alpha_{m2}+\beta_{m2} & \ldots & \alpha_{mn}+\beta_{mn} \cr }.$$
The resulting matrix $C\in M_{m\times n}(F)$ is called a matrix sum of the two matrices \(A\) and \(B\).
Vector addition is a special case of matrix addition for matrices of the size $M_{m\times 1}(F)$ for column vectors. $$ \pmatrix{ \alpha_{1} \cr \alpha_{2} \cr \vdots \cr \alpha_{m} \cr }+\pmatrix{ \beta_{1} \cr \beta_{2} \cr \vdots \cr \beta_{m} \cr }=\pmatrix{ \alpha_{1}+\beta_{1}\cr \alpha_{2}+\beta_{2}\cr \vdots \cr \alpha_{m}+\beta_{m} \cr }.$$
or of the size $M_{1\times n}(F)$ for row vectors.
$$ \pmatrix{\alpha_{1},&\alpha_{2},&\ldots, &\alpha_{n}} + \pmatrix{\beta_{1},&\beta_{2},&\ldots, &\beta_{n}}=\pmatrix{\alpha_{1}+\beta_{1},&\alpha_{2}+\beta_{2},&\ldots,& \alpha_{n}+\beta_{n}}.$$
Corollaries: 1
Examples: 2
Proofs: 3
Propositions: 4
