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Scalar Quantity and Scalar Matrix Multiplication

Matrix Scalar Quantity Multiplication L.Matrix Scalar Quantity Multiplication R.
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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

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

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