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 A − B 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 (3A − B):
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.
- A + B = B + A Commutative Property
- A + (B + C) = (A + B) + C Associative Property
- (cd) A = c (dA) Associative Property Scalar Multiplication
- c (A + B) = cA + cB Distributive Property
- (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 = A − B 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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