 Sitemap

Mobile Math Website

# Scalar Quantity and Matrix Multiplication

Scalar multiplication operations with matrices come from linear algebra where it is used to differentiate a single number from a matrix; that single number is a scalar quantity. Scalar is an important matrix concept. In broader thinking it means that the quantity has only magnitude, no direction.

To understand scalar outside of mathematics think of a speedometer in an automobile that shows the velocity (speed) of that vehicle at any moment. There are millions of automobiles each going a different direction at any given time; the speedometer does not tell us anything about the vehicle direction so it is scalar.

A scalar quantity or multiple scalar quantities are components of a vector. Vectors have both magnitude and direction. Acceleration is a vector quantity. Acceleration is a directional force or multiple different directional forces acting upon an object, the result can be measured as the object moves a particular direction at a speed (its velocity).

In direct terms of scalar multiplication we can multiply a matrix A by a scalar multiple c by multiplying each element in matrix A by c. We can state this mathematically as:

A Formal Definition of Scalar Multiplication

If A = [ ay ] is an m×n matrix and c is a scalar, the scalar multiple of A by c is the m×n matrix given by:

cA = [ cay ]

If we state −A is the negative of matrix A we can also state this as a scalar product, −1×A. Then if A and B both are of the same order then AB represents the sum of A and (−1)B as:

A − B = A + (−1)B

## How to Perform Scalar Multiplication

Multiply matrix A by scalar quantity 3:

Matrix A

3A = 3 ×

 3×3 Column 1 Column 2 Column 3 Row 1: 2 −3 4 Row 2: 6 0 3 Row 3: 7 1 −5

Matrix A With Scalar Multiplier 3

=

 3×3 Column 1 Column 2 Column 3 Row 1: 3(2) 3(−3) 3(4) Row 2: 3(6) 3(0) 3(3) Row 3: 3(7) 3(1) 3(−5)

Result Matrix (3A)

 3×3 Column 1 Column 2 Column 3 Row 1: 6 −9 12 Row 2: 18 0 9 Row 3: 21 3 −15

Multiply matrix B by scalar quantity −1:

Matrix B

−1B = −1 ×

 3×3 Column 1 Column 2 Column 3 Row 1: 0 2 0 Row 2: 3 −3 5 Row 3: −2 4 −1

Matrix B With Scalar Multiplier −1

=

 3×3 Column 1 Column 2 Column 3 Row 1: −1(0) −1(2) −1(0) Row 2: −1(3) −1(−3) −1(5) Row 3: −1(−2) −1(4) −1(−1)

Result Matrix (−1B)

 3×3 Column 1 Column 2 Column 3 Row 1: 0 −2 0 Row 2: −3 3 −5 Row 3: 2 −4 1

Solution to scalar matrix expression
(3AB):

Matrix (3A)

3A − B  =

 3×3 Column 1 Column 2 Column 3 Row 1: 6 −9 12 Row 2: 18 0 9 Row 3: 21 3 −15

Matrix (−1B)

 3×3 Column 1 Column 2 Column 3 Row 1: 0 −2 0 Row 2: −3 3 −5 Row 3: 2 −4 1

Result Matrix (3A B)

 3×3 Column 1 Column 2 Column 3 Row 1: 6 −7 12 Row 2: 21 −3 14 Row 3: 19 7 −16

## Properties of Scalar Addition and Scalar Multiplication

Let A, B and C be m×n matrices and let c and d be scalar quantities.

1. A + B = B + A
Commutative Property
2. A + (B + C) = (A + B) + C
Associative Property
3. (cd) A = c (dA)
Associative Property Scalar Multiplication
4. c (A + B) = cA + cB
Distributive Property
5. (c + d) A = cA + dA
Distributive Property

Scalar Identity Property

1A = A

The number 0 is the matrix additive identity for real numbers.

If we take a scalar and multiply it by a matrix of order m×n where all matrix elements are zeros the result is the scalar value c.

c + 0 = c

In similar thinking, if we take a 0 value scalar c and add it to matrix A of order m×n the resulting matrix is identical to matrix A.

A + 0 = A

## Solving a Matrix Equation

If given the matrix equation 4X = AB we would first factor so that X is isolated to one side of the equation:

X = ¼ (A − B) = ¼ A − ¼ B

Next we would perform scalar multiplication on matrix A and then the scalar multiplication for matrix B.

The final matrix math process step would be to subtract the result matrices A and B. This result gives us matrix X. The matrix equation is solved.