Square Matrix

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A square matrix is popularly known as an nxn matrix that contains an equal number of rows and columns. It is an effective way to analyse, arrange and represent data in a logical structure. The matrix is also used in mathematical equations and can provide an approximation of complicated calculations. Matrices can be of different types, and invertible matrix, symmetric matrix, singular matrix, etc. are among the important ones.


Types of Square Matrices

The following are some of the important types of square matrices. 

  • Invertible Matrix 

It is a kind of square matrix and the product of this type a matrix and its inverse is an identity matrix. The use of this invertible matrix generally is seen in the different fields of science to decrypt any coded message. 


Nevertheless, a unit square matrix is another useful algebraic expression which comes from the transformation or multiplication of matrices. This unit square is basically a square with different vertices (0,0), (0,1), (1,0), (1,1). 

  • Symmetric Matrix 

A symmetric matrix is also a square matrix that follows, A\[^{T}\] = A. 


Here, A\[^{T}\] is transpose f A and A\[^{-T}\] A\[^{-1}\] = I and here I stands for invertible matrix. 

  • Singular Matrix 

One of the vital features of singular matrices is that the determinant of it has to be 0. A matrix that cannot be inversed is also called a singular matrix. For example, if the element of the first column of a matrix is zero, then the determinant is zero as well. 


Singular matrices are not an invertible matrix. This type of matrix works as a barrier between such matrices whose determinants are positive and whose determinants are negative. Solve and practice singular matrix example 3x3 for a better understanding of this concept. Nevertheless, besides knowing about these types of matrices, students also need to know how to find diagonal of a square. 


How to Find the Diagonal of a Square Matrix? 

In order to find the diagonal of a square matrix, firstly, you need to consider the first element in the 1st row and the last element in the last row. Moreover, consider all the elements that are linked in a straight way through a diagonal straight path in the matrix. 


\[\begin{bmatrix} 9 & 0 & 1 & 1\\ 0 & 11 & 1 & 0\\ 1 & 1 & 4 & 1\\ 1 & 0 & 1 & 10 \end{bmatrix}\]


Here, the elements, 9, 11, 4, 10 can be joined diagonally using a straight line. Therefore, these elements are called diagonal of the square matrix. 


How to Square a Matrix?

When we multiply two matrices, it is needed to ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix. For square matrices, students only have to multiply the elements of the two matrices to find the product. To find the square of a matrix, the matrix has to be multiplied by itself. 


Nevertheless, if you want to deeply delve into the core of the invertible matrix concept further, you can refer to the information on Vedantu. Besides these concepts of linear algebra, you can also learn several other concepts by downloading the PDF files of study materials on Vedantu app and website. It also contains the important questions and answers so that you can lay a strong foundation of the basic concepts of algebra. 


Therefore, download the Vedantu App today and keep learning on the go. 

FAQ (Frequently Asked Questions)

1. What is a Unit Matrix of Order 3? 

Ans. The unit matrix or identity matrix of 3 x 3 order is given below. 

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In this matrix, there are three rows and three columns. All the main diagonal elements are 1s in this matrix and rest of the elements are 0s in this unit matrix. 

2. What is Identity Matrix? 

Ans. An identity matrix is a kind of matrix, more specifically n x n matrix where all diagonal elements are 1s, and other elements are 0s. The identity matrix is denoted by I. It also means there are n columns and n rows in this square matrix. 


The role of Identity matrix is similar to that of 1 in the number system of real numbers. Nevertheless, the product of an identity matrix and square matrix comes out as a number. 

3. What is a Matrix Order? 

Ans. The matrix order refers to the number of rows and columns that a matrix has. It is also called dimensions of a matrix. Conventionally, while writing the order of matrices, the number of rows is written first and is followed by the number of columns. For instance, in a matrix of unit 7 x 8, the number of rows and columns is 7 and 8, respectively. Moreover, it also refers that the total number of elements present in the matrix is 7 x 8= 42. 

4. Where is Matrix Used in Real Life?

Ans. Primarily matrices are used in statistics, plotting graphs, conducting scientific studies etc. It is one of the most useful representative methods that is used to showcase data in a more organised way. However, different types of matrix serve different aspects that students need to know while learning those lessons.