> [!summary] Matrix addition and scalar multiplication are two operations assumed to be true when matrices $A,B$ are the same size. > **Key Equations:** > Matrix Addition: $(A+B)_{ij} = A_{ij} +B_{ij}$ > Scalar Multiplication: $(tA)_{ij} = t \cdot A_{ij}$ >[!info]+ Read Time **⏱ 1 min** # Definition [[Matrix Notation|Matrix]] addition is an assumption that two matrices that are the same size can be added. Meaning that each entry can be added to the other. If $A,B$ were to matrices in $\mathbb{R}^{n\times m}$ with entries $a_{ij},b_{ij}$, the sum of the two is $A+B = C$ where $C\in \mathbb{R}^{n\times m}$ defined below $ \begin{array}{c} \\ \text{Entires:} \\ A = [a_{ij}] \\ B = [b_{ij}] \\ C = [c_{ij}] \\ \\ \text{So the sum of } A+B=C \space \text{is defined through C} \\ c_{ij} = a_{ij}+b_{ij} \quad \text{for all} \space 1 \leq i \leq n, \space 1 \leq j \leq m \\ \\ \text{So the sum} \\ (A+B)_{ij} = A_{ij} +B_{ij} \end{array} $ Scalar multiplication is an assumption that a [[Scalar & Vectors|scalar]] [[Real Numbers|real number]] can be multiplied by a matrix. If a $A$ was a [[Matrix Notation|matrix]] in $\mathbb{R}^{n\times m}$ and $t$ was a scalar real number, then the following can be said $ \begin{array}{c} A = [a_{ij}] \in \mathbb{R}^{n\times m}, t\in \mathbb{R} \\ \\ \text{Then tA = B} \\ B = [b_{ij}]\in \mathbb{R}^{n\times m} \\ b_{ij} = t\cdot a_{ij} \quad \text{for all} \space 1 \leq i \leq n, \space 1\leq j\leq m \\ \\ \text{So then:} \\ (tA)_{ij} = t \cdot A_{ij} \end{array} $