Types of matrices

Introduction

Matrices are the plural form of a matrix. Matrices are horizontal and vertical arrangements of numbers, expressions, variables, etc. in a rectangular array. The entries in the array are termed as elements, which are used to perform various operations with each other. There are many certain types of matrices used for different purposes.

So, now, we are going to learn about their types and properties in detail. Matrices are one of the major parts of mathematical operations,

About Matrices

Definition:

Matrices are rectangular-shaped arrays containing entries of elements. There are horizontal and vertical entries in the array. Vertical entries are referred to as columns, whereas horizontal entries are referred to as rows.

Matrixes come in a variety of shapes and sizes, which are governed by their dimensions. A = [aij]pxq is a formula that is commonly used to identify the type of matrices.

Types of matrices:

There are many types of matrices. Matrix types are determined by their dimensions. Let’s discuss their types.

Row matrix:

Row matrix is a matrix in which there is only a single vertical row of elements. Suppose if there is only one row in the matrix, Hence p = 1; so,    p x q = 1 x q. By putting value in general formula, A= [aij]p×q, we get A= [aij]1xq. Row matrices can be represented by this formula. As there is only one row and denotes the number of columns = q.

[Note: p is the number of rows and q is the number of columns in a matrix, taken throughout the article].

E.g. A = 4     1    10    5  in this row matrix, there is one row and four columns. We can conclude that this row matrix is in order of 1 x 4. 

Column matrix:

Column matrix is similar to the row matrix, the matrix in which there is only a single horizontal row of elements is termed a column matrix.

Let there be one column in the matrix, Hence q = 1; so, p x q = p x 1. By applying this in the general formula, we get a column matrix: A= [aij]p×q, we get A= [aij]px1. 

E.g. A =

In this column matrix, there are four rows and one column. By this, we can conclude that this column matrix is in order of 4 x 1. 

Rectangular matrix:

A matrix that doesn’t have the same number of rows as its number of columns is called a rectangular matrix. 

Eg. A =

Square matrix:

A square matrix is a matrix in which all of the dimensions are equal i.e. columns and rows. That forms a square array and hence it is called a square matrix. Thus, A=[aij]p×q can be considered a square matrix if p = q.

Eg.  A=

Diagonal matrix:

A diagonal matrix is always a square matrix. The diagonal matrix is of dimensions n x n, in which elements forming diagonals are only non-zero elements in the matrix other elements are zero or null. Matrix ‘A’ is said to be a diagonal matrix if Apq ≠ 0, and p = q.  

Eg. A= ; A’ =

As all non-diagonal elements are valued 0 in the above examples, we can say that matrix A and A’ are diagonal matrices 

Identity matrix:

Identity matrix is a kind of diagonal matrix in which elements forming diagonals is of value 1. The identity matrix’s other properties are similar to the diagonal matrix. 

E.g. 

Singleton matrix:

The matrix in which there is only one element present is called the singleton matrix. Hence, resulting in only one column and row. A= (aij)pxq can be considered as a singleton matrix if p = q = 1.

Eg: A=4; A’=5

As above-mentioned matrix examples have only one element present in them, matrix A and A’ both are singleton matrices.

Zero or null matrix:

Zero or null matrix has all elements valued 0. Matrix A can be said to be a zero or null matrix if A= (aij)pxq =0 for all p and q.

Eg. A=

This is the example of a null matrix; in this matrix, all elements are valued at zero, and we can say that matrix A is a null or zero matrix.

Equal matrix:

When two matrices have the same number of rows and columns and their elements are corresponding to each other, then, they are said to be equal matrices.

E.g. A= B =

This is an example of equal matrices; in this, the dimensions of both matrices are equal and their elements are corresponding to them as well, so we can say matrices A and B are equal matrices.

Symmetric & Skew symmetric matrix:

A symmetric matrix has the same dimensions as a square matrix. The transpose of systematic matrices is equal to that of real matrices. The symmetric matrix is A=(aij), if for all i,j values, Aij = Aji. This matrix can only be called a symmetric matrix if its dimensional coordinates are of equal value. i.e. a12 = a21, a32 = a23 .

E.g. A= ; AT =

In the example given above, the coordinates of dimensions are corresponding to each other so we can say that matrix A is a symmetric matrix.

On the other hand, a skew-symmetric matrix has the same rules as a symmetric matrix but the difference is, in a skew-symmetric matrix all diagonal elements are valued at zero.

E.g. A= ; A’ =   Here, AT = – A’

In the above example, the diagonal elements are valued at zero, and coordinates also correspond. This matrix can be now considered as a skew-symmetric matrix.

Other matrices

These were some of the main matrices, many matrices have different structures and rules. There are some common matrices among them.   

Horizontal matrix:

The matrix in which rows are more than columns, that matrix can be termed as a horizontal matrix. (p < q)

Eg.  A=

Vertical matrix:

The vertical matrix is the reverse of the horizontal matrix. In the vertical matrix, there are more columns than rows. (p > q)

Eg.  A=

Orthogonal matrix:

The matrix in which its transpose and inverse are equal is called an orthogonal matrix. 

AT=A-1 which further conveys AT A = AAT                     

Conclusion

Today, we got to know about the matrix types, their meanings with formulas, and examples of each type. Matrices are a very important part of algebra, they cover the descent chunk in the syllabus of JEE(Mains) and JEE(Advance). It is necessary for students preparing for such highly competitive exams to prepare on concepts like matrices and other related topics.