Scalar Quantity and Scalar Matrix Multiplication

Matrix Scalar Quantity Multiplication L.Matrix Scalar Quantity Multiplication R.
HomeAlgebraMatrices and DeterminantsMatrix 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 ×

Matrix A

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:

-1B = -1 ×

Matrix B

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):

3A – B  =

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

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

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.

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