A symmetric property is a feature of sameness that can be applied to different sets of expressions. We mean that the logic of distribution is the same for two different numbers.
What is a symmetric property?
A symmetric property is an important concept in Algebra that is applied in so many mathematical problems such as congruence, relations, matrices, equality, and so on.
Generally, the law of symmetric property as related to a set states that when an element is mathematically related to another element, then this second element also has a mathematical relationship with the first element based on the law of relationship that has been defined by the set.
The symmetric property as used in algebra
Symmetric property as used in algebra is described as a property that states that when one element that is contained in a set has a mathematical relationship with a second element, then it is safe to say that this second element also has a relationship with the first element. When it comes to the symmetric property, there are different forms.
Types of symmetric properties
There are four main symmetric properties. These are:
The symmetric property of Matrices
Symmetric property of Relations
Symmetric property of Congruence
Symmetric property of Equality
The symmetric property of Equality
Since we now know what a basic symmetric property is, we can now go on to explain the meaning of the symmetric property of Equality. The law of the symmetric property of equality states that assuming a real number, x is equal in value to another real number, y, then it is right to say that y and x are both equal in value.
This is an important relationship that helps to define a mathematical set of numbers. It is defined as if ‘aRb when only if a=b’ is a symmetric relation. Then mathematically, it is valid for us to show the symmetric property of equality to be ‘If x=y, then y=x’. It is an important property that can also be used to look for the value of different variables on a set of equations.
The symmetric property of congruence
A symmetric property of congruence says that when a geometric figure shares a congruency with another, then it is right to deduce that the second figure is in congruency with the first figure.
For instance, assuming that a triangle, ABC is in congruency with another triangle, PQR, then it is right to say the latter triangle, PQR, is in congruency with the first triangle ABC.
This theory can also be expressed using another logic. Given that AB, a segment of a line is in congruency with another segment of a line, CD, then it is safe to say that the line segment, CD, is in congruency with AB. This important property can be applied to angles as well as various other geometric shapes and figures.
Symmetric property of Matrix
A symmetric property of a matrix has a law which states that given that a certain Matrix, A is symmetrical, this means that A will be equal to the result of its transpose.
This can be written mathematically as A = AT
When it comes to symmetric properties, there are certain properties involved.
They have a commutative property which states that if a matrix, A is symmetrical to another matrix, B, then this means that AB=BA.
The difference and sum of two symmetrical matrices will always result in a symmetric matrix.
For any integer given as n, if a matrix A has a symmetric nature, then it means that An has a symmetrical feature.
For a symmetrical matrix A, this means that A-1 have a symmetrical feature.