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A scalar matrix can simply be defined as a diagonal matrix in which all the elements of the diagonal matrix are equal. A diagonal matrix is a matrix that has non-zero elements diagonally from the upper left corner to the lower right corner. A scalar matrix is a diagonal matrix that has all the other elements as zero except the diagonal one. The others are of the same element. There are a number of matrices for example diagonal matrix, triangular matrix, column matrix, row matrix, and many others. A matrix is of both row and column. Also, matrices form rectangular arrays.
Properties of Scalar matrix:
Listing below are some of the most important properties of the scalar matrix.
K(A+B) = KA+KB
Where K is the Scalar, A, and B is the matrix. Let us understand with the help of an example.
If K= 2
A= [ 4 6 ]
B= [ 6 4 ]
A+B = [ 10 10 ]
*( Adding the matrixes, 4+6 and 6+4)
LHS
K(A+B) = 2(10+10)
K(A+B)= [20 20]
RHS
KA+KB
KA= 2[ 4 6 ] and KB= 2[ 6 4 ]
*Multiplying K and A
KA= [ 8 12 ] and Kb= [12 8 ]
Now,
KA + KB = [ 20 20 ]
* Adding the value of KA and KB
K(A+B)= KA+KA
[ 20 20 ] = [ 20 20 ]
Hence proved LHS=RHS.
(K+L)A = KA+LB
If L=1
If K= 2
A= [ 4 6 ]
B= [ 6 4 ]
LHS
(K+L)A= 2+1[ 4 6 ]
3[ 4 6 ]
[ 12 18 ]
(K+L)A= [12 18 ]
RHS
KA+LB= 2[ 4 6 ]+ 1[ 6 4 ]
[ 8 12 ] + [ 6 4]
[ 12 18 ]
(K+L)A= KA+LB
[12 18 ] = [12 18 ]
Therefore LHS=RHS.
These were two of the most important properties of the scalar matrix.
Matrices.
A matrix can simply be defined as a rectangular array that is formed when a sum of numbers or a set of numbers is formulated in rows and columns. Matrices are basically used to represent a mathematical calculation or mathematical object or the subject of that project.
An example of a matrix would be:
There are a lot of different types of matrices that are involved. In this article, we will be taking a headnote to diagonal matrices and the Scalar matrix.
Diagonal matrix.
A diagonal matrix is that matrix in which all the numbers in the matrix are zero except the one which is diagonal in nature. Only the numbers that are diagonal in the matrix from the top left side to the bottom right side are called the diagonal matrix.
Here is an example of a Diagonal matrix.
Diagonal matrix example.
This is how a diagonal matrix looks.
Scalar matrix
A scalar matrix can simply be defined as one of the many parts of the diagonal matrix. It is that matrix in which all the elements of the matrix are zero except that of the diagonal one. Same as that of the diagonal matrix, but there is a catch to this. A scalar matrix is one in which all the elements of the matrix are zero except the diagonal one but the elements in the diagonal matrix are the same. All the diagonal elements in the scalar matrix necessarily need to be the same element.
Following are some of the scalar matrix examples.
Scalar matrix example.
The diagonal matrix is of the same elements.
There are a lot of ways in which the numbers can be calculated and arranged and the matrix is one of the very famous ways in which the numbers are calculated and arranged. This makes it easier for people to understand. Matrix is a kind of computer language which makes it easier for anyone to follow the calculation and arrangements.
Matrices are of different types: row matrix, coulomb matrix, and Square matrix.
A scalar matrix comes in the form of a Square matrix, as it is important to form a diagonal pattern in terms to make the matrix scalar.
Square Matrix
A square matrix is the combination of both the row matrix and the column matrix. It is combined of rows and columns, mostly in the form of 2*3 or 3*3. The square matrix is the basis of the scalar matrix. Any matrix which is not square in nature can ever be called a scalar matrix.
Here is an example square matrix.
Conclusion.
This was all about the scalar matrix, the properties of the scalar matrix and some of the most common examples of scalar matrices. The scalar matrix cannot be formed without it being a square matrix, every scalar matrix is a square matrix but every square matrix is not a scalar matrix. It is one of the most common and important types of a matrix that is used in day to day life. It is mostly used in statics to formulate/generate scientific research. It is used in day to day life and in the scientific field in more ways than one.
