Symmetric Property

Have you ever noticed how some truths in mathematics work both ways, like a perfect reflection in a mirror? If your friend is standing next to you, then you're standing next to your friend. This beautifully simple idea lies at the heart of one of mathematics' principles known as the symmetric property. 

This concept has several uses in maths. The symmetric property states that if one expression is equal to another, then the other expression is also equal to the first one. This simple rule keeps equality, congruence, and even geometry balanced and logical.

In this blog, we will learn about the symmetric properties of equality, congruence, matrices, and geometry with different examples for better understanding.

What is the Symmetric Property?

The symmetric property states that if one relationship is valid on one side, then it must be valid on the other side. For Example, a=b then b=a. 

Various forms of Symmetric properties are : 

• Symmetric Property of Equality
• Symmetric property of congruence 
• Symmetric Property of Matrices
• Symmetric Property of Geometry

What is a Symmetric Property of Equality?

In simple words, the symmetric property of equality states that: if one quantity is equal or congruent to another, then the reverse is also true. Here are some examples of the symmetric property of equality :  

• Symmetrical Property in Arithmetic 

If    4+2=6then 6=4+2

• Symmetrical property in Algebra

If x=3 then 3=x 

• Symmetrical Property in a Matrix 

If A=ATthe matrix is symmetric

• Symmetrical property in congruence  

If ABC=DEF  then DEF=ABC

This concept is called the symmetric property of equality. It shows that equality works on both sides and makes sure that mathematical statements stay balanced and true.

Symmetrical Property of Geometry

In geometry, the symmetric property means that if one geometric figure is equal to or is congruent to another, then the reverse is also true.

It is used when proving, shapes, angles, and triangles are congruent

For example: 

• If line AB =line CD, then line CD =line AB

Symmetrical Property of Congruence:

The symmetric property of congruence states that if one shape is congruent to another, then the second shape is also congruent to the first one.

For Example:

• If line segment AB≅CD, then CD≅ AB. 
• If another example of triangle ABC is congruent to triangle DEF, then triangle DEF is also congruent to triangle ABC.
• This property helps prove geometric theorems and supports logical reasoning in proofs.

Symmetrical Property of Matrix:

The symmetric property of a matrix states that if a matrix B is symmetric, then it is equal to its transpose.

For example : 

In this Example, there is a matrix B. Now we take the transpose of matrix B, which means changing rows into columns and columns into rows. After taking the transpose, the matrix is equal to B, which means Matrix B is symmetric.

Properties of a Symmetric matrix 

Some key properties of asymmetric matrix are : 

• Symmetric matrices are commutative, which means that if A and B are symmetric matrices, then AB=BA
• The Sum or difference of two symmetric matrices also results in a symmetric matrix.
• The elements across the diagonal are equal.
• If B is symmetric, then B^T also symmetric.
• If B is a symmetric matrix, then B^2 so is also symmetric.

Skew-symmetric Matrix: 

A skew-symmetric matrix states that the Transpose of A is equal to its negative. For example 

In this example, we see that the transpose of B is equal to -B, which satisfies the skew-symmetric property.

Symmetric Matrix Determinant

The determinant of a symmetric matrix states that for a symmetric matrix A, its determinant is equal to the determinant of its transpose, 

•  It does not change the direction only shows how much the matrix stretches, shrinks, or flips its special vectors.

For example 

Step 01: Write the equation 

Eigenvalues() satisfy the equation: 

Which means

Here : 

So, Eigen values are

Properties of the Symmetric matrix determinant:

• The determinant is always real.

Transitive Property

A transitive property states that any value that is equal to the second one, the second value is third the and the first equals the third. It shows a chain relationship between the first and last value.

For example 

Differentiate Symmetric, Reflexive, and Transitive Properties 

 The main differences between the symmetric are: 

The Symmetric property means that if one value is equal to the first, then the second value is equal to the first