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Mathematics · Ch 3 — Matrices

Properties of Scalar Multiplication of a Matrix

3.4.4

Properties of Scalar Multiplication of a Matrix

3.4.4 Properties of Scalar Multiplication of a Matrix

Scalar multiplication of a matrix follows algebraic laws that mirror the properties of scalar multiplication in ordinary arithmetic. Let A=[aij]A = [a_{ij}] and B=[bij]B = [b_{ij}] be two matrices of the same order m×nm \times n, and let kk and ll be scalars (real numbers). The following three properties are the fundamental laws.

Property (i): Distributivity of Scalar Multiplication over Matrix Addition

k(A+B)=kA+kBk(A + B) = kA + kB

Statement: A scalar multiplying the sum of two matrices gives the same result as multiplying each matrix by the scalar and then adding.

Proof: Working element by element,

k(A+B)=k([aij]+[bij])=k([aij+bij])(definition of matrix addition)=[k(aij+bij)](definition of scalar multiplication)=[kaij+kbij](distributive law for real numbers)=[kaij]+[kbij]=k[aij]+k[bij]=kA+kB.\begin{aligned} k(A + B) &= k\left([a_{ij}] + [b_{ij}]\right) \\ &= k\left([a_{ij} + b_{ij}]\right) \quad \text{(definition of matrix addition)} \\ &= \left[k(a_{ij} + b_{ij})\right] \quad \text{(definition of scalar multiplication)} \\ &= \left[k a_{ij} + k b_{ij}\right] \quad \text{(distributive law for real numbers)} \\ &= [k a_{ij}] + [k b_{ij}] = k[a_{ij}] + k[b_{ij}] = kA + kB. \end{aligned}

Note

This is the left distributive law — the scalar multiplies the whole sum on the left. No separate right distributive law is needed, because a scalar commutes with matrix entries and can be placed on either side of the matrix.

Property (ii): Distributivity of Scalar Addition over a Matrix

(k+l)A=kA+lA(k + l)A = kA + lA

Statement: The sum of two scalars multiplying a matrix gives the same result as multiplying the matrix by each scalar separately and adding.

Proof:

(k+l)A=(k+l)[aij]=[(k+l)aij](definition of scalar multiplication)=[kaij+laij](distributive law for real numbers)=[kaij]+[laij]=k[aij]+l[aij]=kA+lA.\begin{aligned} (k + l)A &= (k + l)[a_{ij}] \\ &= \left[(k + l)a_{ij}\right] \quad \text{(definition of scalar multiplication)} \\ &= \left[k a_{ij} + l a_{ij}\right] \quad \text{(distributive law for real numbers)} \\ &= [k a_{ij}] + [l a_{ij}] = k[a_{ij}] + l[a_{ij}] = kA + lA. \end{aligned}

Tip

This is useful when you need to split a scalar factor, e.g. 3A=(2+1)A=2A+A3A = (2+1)A = 2A + A.

Property (iii): Associativity of Scalar Multiplication

k(lA)=(kl)Ak(lA) = (kl)A

Statement: Multiplying a matrix by ll and then by kk gives the same result as multiplying directly by the product klkl.

Proof:

k(lA)=k(l[aij])=k([laij])(definition of scalar multiplication)=[k(laij)]=[(kl)aij](associative law for real numbers)=(kl)[aij]=(kl)A.\begin{aligned} k(lA) &= k\left(l[a_{ij}]\right) = k\left([l a_{ij}]\right) \quad \text{(definition of scalar multiplication)} \\ &= \left[k(l a_{ij})\right] = \left[(kl)a_{ij}\right] \quad \text{(associative law for real numbers)} \\ &= (kl)[a_{ij}] = (kl)A. \end{aligned}

Additional Important Results

These consequences of the above are also used frequently:

  • Scalar 1: 1⋅A=A1 \cdot A = A, since 1⋅[aij]=[aij]1 \cdot [a_{ij}] = [a_{ij}]. …
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