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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:

 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

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)

Matrix A With Scalar Multiplier 3

=

3×3

Column 1

Column 2

Column 3

Row 1:

6

-9

12

Row 2:

18

0

9

Row 3:

21

3

-15

Result Matrix (3A)

Multiply matrix B by scalar quantity -1:

 -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

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)

Matrix B With Scalar Multiplier -1

=

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 (-1B)

Solution to scalar matrix expression
 (3AB):

3×3

Column 1

Column 2

Column 3

Row 1:

6

-9

12

Row 2:

18

0

9

Row 3:

21

3

-15

Matrix (3A)

 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

3×3

Column 1

Column 2

Column 3

Row 1:

0

-2

0

Row 2:

-3

3

-5

Row 3:

2

-4

1

Matrix (-1B)

Result Matrix (3A B)

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

Matrix Additive Identity

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 be to subtract the result matrices A and B. This result gives us matrix X. The matrix equation is solved.