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Current Location > Math Formulas > Linear Algebra > Scalar Multiplication

Scalar Multiplication

The matrix obtained by multiplying every element of a matrix A by a scalar λ is called the scalar multiple of A by λ.
For example: If

Then the product of 3A will be:

Properties of Scalar Multiplication:
All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalars.

· If A and B be two matrices of the same order and if k be a scalar, then:
k (A + B) = kA + kB

· If k₁ and k₂ are two scalars and if A is a matrix, then:
(k₁ + k₂)A = k₁A + k₂A and k₁ (k₂A) = k₂(k₁A)

Example 1: Let

, find the product of 4A.

Example 2: Find out whether or not c (Ax) = A (cx) is a valid equation, where c is a scalar, A is a 2 by 2 matrix, and x is a dimension 2 column vector?

Answer: Yes, c (Ax) = A (cx) is a valid equation, because the order of multiplication does not matter for real numbers.

If

and the scalar c = c, we have:

and

We can see that c(Ax) and A(cx) both produces the same matrix at the end. Hence c (Ax) = A (cx).
Note: c (Ax) = A (cx) holds true for all matrices that have the correct dimensions so that the product of Ax is valid.

Example 3: Find the product of

.

Solution: