Scalar multiplication of matrices

Definition

Let $R$ be a ring, let $c \in R$, and let $A = (a_{ij})$ be an $m \times n$ matrix over $R$. The scalar multiple $cA$ is the $m \times n$ matrix with entries $$(cA)_{ij} = c\, a_{ij}$$ for $1 \le i \le m$ and $1 \le j \le n$.

Remarks

Over a field $F$, matrix addition and scalar multiplication make $M_{m \times n}(F)$ a vector space of dimension $mn$. The rules $c(A + B) = cA + cB$, $(c + d)A = cA + dA$, $(cd)A = c(dA)$ and $1A = A$ hold. For example, $2 \begin{pmatrix} 1 & -1 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -2 \\ 0 & 6 \end{pmatrix}$.

Explanations

L1 human

Multiplying a matrix by a number means multiplying every entry by that number. The size of the matrix does not change. Doubling a matrix doubles every number in the table.