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Scalar multiplication

Created over 8 years ago, updated 10 days ago

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.