Properties of scalar multiplication of a matrix

A matrix is a rectangular array of numbers with columns and rows separated. A scalar is a real number in matrix algebra. When we work with matrices, we consider the real numbers as scalars. The scalar multiplication and the matrices operation refer to the actual product of numbers and a matrix. While in the scalar multiplication, the entry in the matrix is multiplied by all the given scalar. A matrix is a rectangular arrangement of columns and rows. Each number in the matrix we solve is considered to be the matrix element or entry. Check out the further details if you are new to know the further details of the matrices. 

Matrices and scalar multiplication

When working with matrices, real numbers are referred to as scalars. Scalar multiplication is the process of multiplying a real integer by a matrix. In scalar multiplication, every entry from the matrix is multiplied by the supplied scalar. Because scalar multiplication is substantially based on real-number multiplication, many of the multiplication properties that we know apply to real number multiplication apply to scalar multiplication as well.

Because scalar multiplication is substantially based on real-number multiplication, many of the multiplication properties that we are familiar with in real-number multiplication also apply to scalar multiplication. X and Y are equal-dimensional matrices, c and d are scalars, and O is a zero matrix in the table below.

The explanations for these features are listed below.

We can explain the following scalar multiplication qualities if we create two matrices of arbitrary order as X and Y and then define c and d as a scalar:

1. Closure property of multiplication

A scalar multiple of a matrix is, in most cases, another matrix of the same dimension. The closure property of scalar multiplication refers to this.

When a matrix is multiplied by the actual scalar, a new matrix always has the exact dimensions as the matrix in the multiplication. For instance, if you multiply c. X, the resulting matrix has certain dimensions of X.

When you look forward to this, the thing that happens in a scalar multiplication matrix is that each of the components is multiplied by the scalar outside; and further, there is no extra feature. Thus the elements stay put, and the output is a matrix of the same and actual size.

2. Commutative property

The order of the factors is put in the operation certainly makes no impact when multiplying a matrix times a scalar. In other words, no matter how we order the processes, the outcome of scalar and the matrix multiplication of c and X or d and Y remains the same ie.,c. X = X . c      

3. Associative property of multiplication

If a matrix is multiplied by two scalars, this characteristic states that you can multiply the scalars together first, then multiply by the matrix. Alternatively, you can multiply the matrix by one scalar and then multiply the resulting matrix by the other scalar.

Because of the associative characteristic, scalar multiplication can be done in phases. This property states that even if there are many factors involved, such as many scalars times matrix, you can choose to multiply two factors first and then further use the result of that operation to multiply for another factor that hasn’t been multiplied yet, and so on until you’ve finished multiplying all of the certain factors in the multiplication to get the result. 

Let’s look at an example: suppose you’re multiplying scalars c and d by matrix X. Instead of doing the two multiplications at the same time, the associative property allows you to choose two of these components (c, d, and X) and multiply them first, then multiply the result by the remaining factor to get the result (When more than three components are present, things might get complicated.). A scalar is a real number used in matrix algebra: 

c . (dX) = (cd)X   

4. Distributive property 

When multiplication of a matrix is used in conjunction with another mathematical operation, preferably like addition or subtraction, the distributive property emerges. In another way, we employ this property to make problems easier when one of the factors in a matrix multiplication is an addition or subtraction. 

This property defines, mathematically, that if one of the elements in the multiplication is the adding up of two other matrices, then: c . (X+Y) = cX + cY 

Conversely, this property states that (c+d) X = cX + dX, if a scalar addition is one of the components in the multiplication.

 5.  Multiplication property for the zero matrices

This property asserts that the result of a multiplying by the zero matrices is always the zero matrices itself as long as a result is defined, that is, as long as all of the essential dimension constraints are met, regardless of whether it is the product of a scalar and a matrix or the product of two matrices.

Because all of the elements of a zero matrix are actual zeros, the scalar multiplication matrix is zero doesn’t matter which scalar is multiplied. They will end up with all entries equal to the outcomes of a zero multiplication, which is zero. As a result, multiplication by the zero matrices always results in a zero matrix. 

As a result, if we do scalar multiplication on matrix 0, we get 

c. 0 = 0 

If you multiply the zero matrices by another matrix, the result is:

0. X = 0 

6. Multiplicative identity property

According to this condition, when you multiply any matrix X by the scalar 1, the result is simply the original matrix X.

Because 1. c = c for every real integer c, the scalar 1 in scalar multiplication will always be the multiplicative identity. 

Conclusion

Given that scalar multiplication of vectors is a relatively simple operation. Scalar multiplication is one of the most straightforward, if not the most straightforward, of all external operations on matrices. The scalar matrix is a square matrix and has a constant value of all the elements along with the diagonal. and other elements of the matrix are found to be zero. The order of the scalar matrix is n x n, where n is the number of rows and number of columns. The matrix is the special kind of diagonal matrix in which the diagonal consists of the same element and has the same value as 1.