Scalar Multiple Property

The scalar multiple properties are the property of vectors and matrices. The multiple scalar property of a v x w matrix and a k x l matrix is the statement that if,

A = UV, B = wx, C = zy 

and D = αzβxγ 

then A × B × C × D must be equal to αuβwγ. In mathematics, the scalar multiple properties result from matrix theory stating that the product of any two matrices whose dimensions are such that one has the rank greater than or equal to 2 and less than or equal to 3 is a scalar matrix.

Matrix:

A matrix is a rectangular array of mathematical objects that consists of rows and columns. Matrices are used to represent linear transformations, such as rotation or translations.

Scalar:

Scalars are real numbers, but unlike vectors and matrices, they have only one dimension (a scalar number has a single unit), something like 

6 or 15 is written with a single character letter in mathematics, not with an expression like “6×5”. Scalars can be combined into vectors and tensors.

Scalar matrix:

A “scalar matrix” is a matrix whose elements are scalars.

Multiplication of scalar matrices:

Two scalar matrices can be multiplied; however, the resulting is not a matrix. The following reasons are given to explain it:

Scalar product:

Let “A” and “B” be two scalar matrices. Their scalar product is given by 

where: 

“a” and “b” are the elements of the two matrices, and 

is a scalar operation on vectors.

Addition of scalar matrices:

Scalar matrices cannot be added because the resulting would be a number, and numbers do not have rows and columns.

Division of scalar matrices:

The division of scalar matrices cannot be performed because it is not a matrix. 

The result is that numbers do not have rows and columns.

Matrix multiplication:

“matrix multiplication” means multiplying a scalar “matrix” by another scalar “matrix.” 

This means multiplying two matrices of the same dimension if the dimensions are compatible.

Scalar multi-product: 

A scalar multi-product is a mathematical product that consists of only one element in mathematics.

Scalar Multiple Property: 

The Scalar multiple properties express some important relationships among the elements of a matrix. If the elements in the “n” row and “m” column are multiplied by a single number, the resulting product equals those elements times that number. So if the element in column “i” is multiplied by the element in row “j,” then the result will be equal to “i*j.” But if this product is multiplied by a number, it will result in a scalar, so instead of writing something like “a*b*c,” we could write “a” or “b,” or even a vector of that magnitude. For example, this property can calculate the product of many numbers all at once or find vector cross products.

Example: Take a matrix A, with two rows and three columns. If the elements in the second row and second column are multiplied by a single number, then the resulting product equals those elements times that number. So if the element in column “i” is multiplied by the element in row “j,” then the result will be equal to “i*j.” But if this product is multiplied by a number, it will result in a scalar, so instead of writing something like “a*b*c,” we could write “a” or “b,” or even a vector of that magnitude. For example, this property can calculate the product of many numbers all at once or find vector cross products.

Example: Take a matrix A, with two rows and three columns. If the elements in the second row and second column are multiplied by a number “k,” the result will be equal to “k*a.” If we multiply this product by the value of the first row and first column, we get the correct value, “ka.” So instead of writing something like “a*b*c,” we could write something like “k” or even a vector magnitude since all elements are multiplied by that number.

Example: 

A = [1 2 3 4; 5 6 7 8; 9 10 11 12] and B = [13 14 15 16].

To multiply them together, we will use the scalar multiple properties since, theoretically, it does not matter where we mark our X in the matrices (since columns don’t have numbers, we can easily mark one). So we could choose either A or B.

Conclusion:

This article explained all fundamental concepts about scalar matrices and scalar multiplication. We learned how to deal with matrices and their products, multiply two matrices of different dimensions, and multiply scalar matrices. We used the scalar multi-product, a mathematical product that consists of only one element. 

In mathematics, we also used “scalar multiple properties,” which expresses some important relationships among the elements in a matrix. It was explained that if the elements on row “e” are multiplied by a single number, then the resulting product is equal to those elements times that number.