Scalar Quantity and Scalar 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 ×

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

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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