Hello mathematics enthusiast, eager to find interesting and informative content from the world of mathematics? Well, you have landed on the right page. Here, we make sure to feed the enthusiastic brains of our readers with engaging content. Today’s topic is matrices. We will read everything about the matrices from their origin, and definition to their classification. Besides that, we will read the purpose and properties of the symmetric matrix, one of the most important matrices of all with wide applications. Therefore you must read the article thoroughly and have interesting takeaways. 

Understanding The Defination & Classification Of  Matrices In Mathematics

Begin by reading the following points as they describe the origin, definition, and classification of the matrices. 

Definition

• James Joseph Sylvester introduced the word “matrix” in the year 1850.
• He described a matrix as an object that gives rise to several determinants or minors, that is to say, determinants of smaller matrices that are derived by removing columns and rows from the original one. 
• A matrix can be represented as a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. 
• They can be used to compactly write and work with multiple linear equations, that is, a system of linear equations. 
• There are various types of matrixes categorized on the grounds of the value of elements, order, number of rows and columns, etc.

Classifications 

• Square Matrix – a square matrix is a kind where the number of rows is equal to the number of columns.
• Identity Matrix – kind of square matrix that does not change any vector after we multiply that vector by that matrix.
• Diagonal Matrix – the diagonal matrix is not necessarily a square matrix one where values outside of the main diagonal will have a zero value, meanwhile, the main diagonal is taken from the top left of the matrix to the bottom right.
• Triangular Matrix – has all values in the upper-right or lower-left of the matrix with the remaining elements filled with zero values.
• Symmetric Matrix – a type of square matrix where the top-right triangle is similar to the bottom-left triangle.
• Orthogonal Matrix – a type of square matrix whose rows are mutually orthonormal and whose columns are mutually orthonormal.

Understanding The Purpose Of Symmetric Matrix & Its Properties 

      From the points below you will get to understand the purpose and properties of a symmetric matrix in mathematics. Make sure that you read all the points carefully to understand the topic well.

Purpose

• When it comes to the study of linear algebra, a symmetric matrix is described as a square matrix that is equivalent to its transpose matrix. 
• Now, a transpose matrix can be defined as any given matrix A that can be presented as A^T. A symmetric matrix A, therefore, fulfills the condition, A = A^T. 
• Among all the various kinds of matrices we read, symmetric matrices are one of the most significant ones that are utilized widely in the field of learning machines and equipment.

Properties

• Suppose we are given two symmetric matrices named A and B, then AB is symmetric if and only if A and B follow the commutative property of multiplication, i.e. if AB = BA.
• In the case of an integer n, if A is symmetric, then, An is symmetric.
• Suppose if you see a representation like  A^-1, it will be symmetric if and only if A is symmetric.
• When we take the sum and difference of any two symmetric matrices it gives us the resultant a symmetric matrix.
• The property mentioned in the above point is not permanently true for the product.