Scalar Multiplication of a Matrix
Scalar multiplication of a matrix multiplies every entry of the matrix by a single number called a scalar.
Formula
(cA)_{ij} = c \cdot a_{ij}
Definition
Scalar multiplication means multiplying a matrix by a single number, the scalar, so every entry inside the matrix gets multiplied by that scalar. For a scalar $c$ and an $m \times n$ matrix $A$, the product $cA$ has entries $(cA)_{ij} = c \cdot a_{ij}$, satisfying $c(A+B) = cA + cB$, $(c+d)A = cA + dA$, $c(dA) = (cd)A$, and $1 \cdot A = A$; together with matrix addition, this gives $M_{m,n}(F)$ the structure of a vector space. More deeply, scalar multiplication endows $M_{m,n}(F)$ with an $F$-module structure, and for square matrices it is central: $cA = Ac$ for all matrices, so the center of $M_n(F)$ contains all scalar matrices $\{cI : c \in F\}$; for $F = \mathbb{C}$, the center is exactly the scalar matrices, by Schur's lemma applied to the irreducible representation of $M_n(\mathbb{C})$ on $\mathbb{C}^n$.
Example
$3 \cdot \begin{bmatrix}2 & -1\\0 & 4\end{bmatrix} = \begin{bmatrix}6 & -3\\0 & 12\end{bmatrix}$, every entry multiplied by $3$. For $A = \begin{bmatrix}3 & 6\\9 & 12\end{bmatrix}$: $(1/3)A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ and $(-1)A = \begin{bmatrix}-3 & -6\\-9 & -12\end{bmatrix}$, scaling the matrix without changing its structure (rank, zero pattern, and so on). The scalar matrix $cI_n$ commutes with every matrix in $M_n(F)$, $(cI)A = c(IA) = cA = c(AI) = A(cI)$, the algebraic content of Schur's lemma for irreducible representations.
Key Insight
Multiplying a matrix by a scalar scales every entry uniformly, like zooming a photograph in or out where every pixel changes by the same factor, and scalar multiplication together with matrix addition gives $M_{m,n}$ the structure of a vector space, with matrices as the "vectors." Schur's lemma in representation theory states that intertwiners of irreducible representations are scalar multiples of the identity, the generalization of the commutant of $M_n(\mathbb{C})$ being scalar matrices, with far-reaching consequences in group representation theory.