Scalar multiplication

**Definition:** The **scalar multiplication** of a matrix $A = (a_{ij})$ by a scalar $c$ is the matrix $cA$ defined by:

$$(cA)_{ij} = c \cdot a_{ij}$$

Every entry of the matrix is multiplied by the scalar.

**Example:**
$$3 \cdot \begin{pmatrix} 1 & 2 \\ 4 & 0 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 12 & 0 \end{pmatrix}$$

**Properties:**
- Distributive over matrix addition: $c(A + B) = cA + cB$
- Distributive over scalar addition: $(c + d)A = cA + dA$
- Associative: $(cd)A = c(dA)$
- Identity: $1 \cdot A = A$
- Zero: $0 \cdot A = 0$

Together with addition, scalar multiplication makes matrices into a **vector space** over the field of scalars.Copy