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Home > News & Articles > Scalar Matrix: Definitions, Examples, Formula, Properties, Terms Related, Matrix Operations, and Applications
Updated on 21st September, 2023 , 5 min read
A scalar matrix is a square matrix in linear algebra with all of the elements on the diagonal having the same scalar value and all of the off-diagonal components being 0. Scalar matrices are important in linear algebra and have various fascinating characteristics and applications that make them worthwhile to study.
A matrix is a rectangular array of integers that is organized into rows and columns. A matrix's size is determined by the number of rows and columns it contains. When a matrix has "m" rows and "n" columns and is expressed as a "m n" matrix, it is said to be a "m by n" matrix. For example, if a matrix has three rows and four columns, the matrix order is "3 x 4." Matrix kinds include rectangle, square, triangular, symmetric, singular, and so on.
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A scalar matrix is a square matrix with all of the primary diagonal elements equal and all of the remaining elements 0. It is a type of diagonal matrix that may be produced by multiplying an identity matrix by a constant numeric value. The below image explains the following matrix is a scalar matrix of order "4 x 4." We can see that all of its primary diagonal components are the same, whereas the remainder are zeros.
When an identity matrix is multiplied by a constant numeric value, a scalar matrix is generated.
Here are some scalar matrix examples-
The following is the formula of the scalar matrix-
The following are some of the properties of a scalar matrix-
The words listed below are some of the most crucial in comprehending the notion of scalar matrix, which are as follows-
This is a straightforward numeric value that can be an integer, rational number, decimal number, or root value. To obtain the scalar matrix, multiply the identity matrix by a constant number. A matrix multiplied by a constant value multiplies with each of the matrix's members.
The diagonal matrix is also a square matrix with elements of varying values across the primary diagonal and zero for all other components. Furthermore, if all of the diagonal elements of the diagonal matrix are made equal, it is referred to as a scalar matrix.
The identity matrix is a square matrix with a multiplicative identity. The diagonal member of the identity matrix is 1, while all other elements are zero. The identity matrix has several uses in matrix multiplication and determining the inverse of a matrix. The identity matrix produces a scalar matrix when multiplied by a constant value.
If the components from the first element of the first row to the final element of the last row are connected by a straight line, all the elements in the matrix that lie on this imaginary straight line form the primary diagonal. The primary diagonal members of an identity matrix are all equal to 1, whereas the principal diagonal elements of a scalar matrix are all equal to a constant value.
The scalar matrix's operations are nearly identical to the arithmetic operations of any other form of matrix. The addition and subtraction of a scalar matrix and any other matrix is the same as the addition and subtraction of any two other matrices. However, below is the multiplication of a scalar matrix by another matrix-
Thus, multiplying a scalar matrix by any other matrix equals multiplying the constant element of the scalar matrix by all the components of the other matrix.
The following are some of the applications of the scalar matrix-
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By - Nikita Parmar 2023-08-25 09:48:44 , 11 min readAns. A scalar matrix is a square matrix with equal primary diagonal members and zeros for the remaining matrix components. We know that an identity matrix is a square matrix with one major diagonal element and zero secondary diagonal elements. A scalar matrix is an identity matrix or a unit matrix.
Ans. A scalar matrix is a square matrix with all of the primary diagonal elements equal and all of the remaining elements 0.
Ans. A zero matrix is one with all of its elements equal to zero. As a result, it is not a scalar matrix.
Ans. The scalar matrix has an order of n x n, where n is the number of rows and columns.
Ans. Although all scalar matrices are diagonal, not all diagonal matrices are scalar. This is due to the fact that a diagonal matrix might have various elements, but a scalar matrix always has the same components.
Ans. A null matrix, sometimes known as a zero matrix, is a square matrix with all of its members equal to zero. As a result, we cannot conclude that it is not a scalar matrix.
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