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

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 |

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

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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