What are the matrices and their types ?

Why do we use linear algebra?

What are the matrices and their types ? Linear Algebra is the branch of mathematics in which we study about the all the physical systems that can be modeled as linear systems. Linear algebra deals with solving the linear system of equations that are developed from some chemical reactions or road map or from component manufacturing company. The word “algebra” is derived from the word “aljabar” meaning broken parts.

What are the matrices?

The matrices are the key unit or building of linear algebra. The general expression for matrix is

A=[aij ]m×n

where ‘a’ represent the entry of the matrix and ‘i&j’ are the subscript which shows the corresponding row and column of the matrix. A matrix can have any number of rows and columns which determine the order of the matrix. Here there are m number of rows and n number of columns and mXn define the order of the matrix.

What are the vectors?

Vectors are the special case of matrix. If there are m number of rows and a single column, then we say it as “column vector” . If there is only one one row and n number of columns then it is called “row vector”. 

The following example illustrates the example of row vector and column vector.

row and column vectors
row and column vectors

Types of matrices

There are various matrices and each of them has its specific characteristics. They are described below with an example.

Square matrix

If a matrix has equal number of rows and columns or if m=n, then such matrices are called the square matrices

square matrix
square matrix

Rectangular matrix

If number of rows and columns of a matrix are not equal then such matrices are called rectangular matrices.

rectangular matrix
rectangular matrix

Symmetric matrix

if AT=A then such type of a matrix is called a symmetric matrix. The following example shows a symmetric matrix

symmetric matrix
symmetric matrix

 

Skew Symmetric matrix

If transpose of A is equal to negative of A i.e.,  AT=-A then such matrix is called skew symmetric matrix.

skew symmetric matrix
skew symmetric matrix

Upper triangular matrix

If a matrix has non zero entries above and on the main diagonal, then it is called an upper triangular matrix.

upper triangular matrix
upper triangular matrix

 

Lower triangular matrix

If all the entries of a matrix below and on the main diagonal are non zero, then we have a lower triangular matrix.

lower triangular matrix
lower triangular matrix

 

Diagonal matrix

If a matrix has only non-zero entries on the main diagonal then it will be called a diagonal matrix.

diagonal matrix
diagonal matrix

In linear algebra we might encounter with any of these matrices.

Scalar matrices

These are the diagonal matrices whose values are identical to each other. Consider the following matrix:

scalar matrix
scalar matrix

In this matrix, the scalar is 3.

The scalar matrices satisfy the following property

If S is any scalar matrix with scalar value= 2 and A is any other matrix then

AS=SA=cA.

For this consider the following example

scalar matrix property.
scalar matrix property.

 

Also read here:

https://eevibes.com/how-to-define-and-solve-system-of-linear-equations-in-linear-algebra/

How to solve system of linear equations in Linear Algebra?

One thought on “What are the matrices and their types ?”

Leave a Reply

Your email address will not be published.