Matrix: Definition, Order of a Matrix, Types and Trace

The knowledge of matrices is necessary in various branches of mathematics.It simplifies our work to a great extent when compared with the other mathematical methods like a system of linear equations in three variables can be solved easily by using the matrix approach.

Matrix is not only used in branches of mathematics and science, but also in genetics, economics, and modern psychology. The result of an experiment can be analyzed easily if represented mathematically by matrices. Now, let us understand some important  basic concepts of Matrices. 

Table of Contents:

• Definition of matrix
• Order of a matrix
• Types of matrices
• Trace of a Matrix
• Properties of Trace of Matrix
• Practice Problems
• FAQs

Definition of Matrix

A matrix is a rectangular arrangement(array) of numbers, variables, or expressions in the form of rows and columns.The numbers or variables inserted are called elements or entries of the matrix.Matrix is denoted by the capital letters while elements of the matrix by small letters.

A matrix is alway enclosed within [  ] or (.).Lets understand this with some examples,

Concept Video: 

Concept of the Day | Matrices | Class 11 & 12 MATHS | JEE 2021/2022 l Keshav Sir | BYJU'S JEE

Order of a Matrix

If a matrix has M rows and n columns then the order of matrix is $m\times n$ ,we read it as m by n.

Let's consider a $m\times n$ matrix 

where $a_{ij}$ denotes the element of $i^{th}$ row and $j^{th}$ column .The above matrix can be denoted denoted as $A={\left[a_{ij}\right]}_{m\times n}$.

Example:

Order of matrix $=3\times 2$,    Here, $a_{11}=2$, $a_{12}=3$,${ a}_{21}=-4,a_{22}=7{,a}_{31}=0,a_{32}=1$

Types of Matrices

• Square matrix: A matrix having same number of rows and columns is called a square matrix.It is denoted as $A={\left[a_{ij}\right]}_{n✕n}$or ${\left[a_{ij}\right]}_n$,we can say that order of A is n. 

Example: 

In any square matrix, the elements $a_{i}j,$, for which $i = j$, then that element  is said to be the principal diagonal element of the matrix. In general, $a_{11},a_{22,}{a}_{33,}\dots \dots ..a_{ii}$  are the principal diagonal elements of a matrix.The diagonal containing principal diagonal elements is known as Principal diagonal of the matrix. Example:

For  

   , Here 2 and 1 are principal diagonal elements

• Rectangular matrix: A matrix having a different number of rows and columns is called a Rectangular matrix  i.e $A={\left[a_{ij}\right]}_{m\times n}$ where $m\ne n$

Example :

$3\times 2$

• Row matrix: A matrix having only one row is called a row matrix or row vector.

Example: $B={\left[1 2 3\right]}_{1\times 3}$

• Column matrix: A matrix which has only one column is called a column matrix

Example: 

• Zero matrix: If all elements of a matrix are zero,then that matrix is called zero matrix or Null matrix. A zero matrix can be a square matrix or a rectangular matrix.

$A={\left[a_{ij}\right]}_{m\times n}$      if $a_{ij}=0$ $\forall$ i and j 

Example:

• Diagonal matrix: A square matrix ${\left[a_{ij}\right]}_n$ is said to be a diagonal matrix if $a_{ij}$ $=0$,$\forall i \ne j.$

Diagonal matrix is represented as $diag\left(a_{11},a_{22}\dots ..a_{nn}\right)$

Example:

       or $L=diag\left(\mathrm{1,0}\right)$   ,$T=diag\left(1,-\mathrm{3,8}\right)$

• Scalar Matrix: A diagonal matrix whose principal diagonal elements are all equal

is called a Scalar matrix.

Example:

• Unit Matrix (Identity matrix): A scalar matrix with principal diagonal entries as 1 is called an Unit or Identity  matrix.An Identity matrix of order n is represented as i_n

Example:

• Upper Triangular Matrix: If all of the elements below the principal diagonal are zero, then the square matrix  ${\left[a_{ij}\right]}_n$ is said to be an Upper Triangular Matrix..

$A=$ ${\left[a_{ij}\right]}_n$ such that $a_{ij}=0$   $\forall i > j$

Example:

• Lower Triangular Matrix: If all of the elements above  the principal diagonal are zero, then the square matrix${\left[a_{ij}\right]}_n$ is said to be a Lower Triangular Matrix.

$A=$ ${\left[a_{ij}\right]}_n$ such that $a_{i}j=0$  , $\forall i < j$

Example:

Trace of a Matrix

Trace of a matrix is the sum of all elements of the principal diagonal of a square matrix.

Trace i.e.  $Tr\left(A\right)=a_{11}+a_{22}+a_{33}+\dots \dots +a_{ii}$

Example: 

 $Tr\left(A\right)=a_{11}+a_{22}=2+1$ $=3$

Properties of Trace of Matrix

Let us consider two square matrices A and B then,

• $Tr\left(KA\right)=K Tr\left(A\right)$  where K is a scalar.
• $Tr\left(A\pm B\right)=Tr\left(A\right)\pm Tr\left(B\right)$
• $Tr\left(AB\right)=Tr\left(BA\right)$
• $Tr\left(ABC\right)=Tr\left(CAB\right)=Tr\left(BCA\right)$
• $Tr\left(A^T\right)=Tr\left(A\right)$

Practice Problems of Matrix

Example: Construct a $2\times 3$  matrix ,whose elements are given by ${\mathit{a}}_{\mathit{i}\mathit{j}}=\frac{\left(\mathit{i}+2\mathit{j}\right)}{2}$ .

Answer:

Given, ${\mathit{a}}_{\mathit{i}\mathit{j}}=\frac{\left(\mathit{i}+2\mathit{j}\right)}{2}$

${\mathit{a}}_{11}=$ $\frac{1+2\times 1}{2}$ = $\frac{3}{2}$  ,${\mathit{a}}_{12}=$ $\frac{1+2\times 2}{2}$ = $\frac{5}{2}$ ,${\mathit{a}}_{13}=$ $\frac{1+2\times 3}{2}$ $=\mathit{ }\frac{7}{2}$ 

${\mathit{a}}_{21}=$ $\frac{2+2\times 1}{2}$ $=2$ ,${\mathit{a}}_{22}=$ $\frac{2+2\times 2}{2}$ = 3 ,${\mathit{a}}_{23}=$ $\frac{2+2\times 3}{2}$ = 4 

∴ Required matrix is 

Example: Which of the following statements is always true if matrix A has n number of elements and m is the number of potential orders for matrix A?

a) $m=1$,when n is prime                          b)  $m=2$ ,when n is prime
 c) m < 2 ,when n is not prime                       d) m > 2 ,when n is prime

Answer:

Matrix A consists of n number of elements

Case$\left(i\right)$ If n is a prime number it can have only two factors 1 and n.

∴ The possible orders are $1\times n$ or $n\times 1$.

Hence, if n is a prime number, $m=2$

Case$\left(ii\right)$ If n is not prime,then we get more than two factors for n

Hence, if n is not prime, $m>2$

Therefore the correct option is (b)

Example: Let
 and
 are two matrices such that their trace is equal,then find the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Answer:

Given $Tr\left(A\right)=Tr\left(B\right)$

$\Rightarrow$ $a^2+b^2+c^2=2a+2b+2c-3$

$\Rightarrow a^2+b^2+c^2-2a-2b-2c+3=0$

Rearrange the terms

$\Rightarrow$ $a^2+1-2a+b^2+1-2b+c^2-2c+1=0$

$\Rightarrow$ ${\left(a-1\right)}^2+{\left(b-1\right)}^2+{\left(c-1\right)}^2=0$

Sum of squares of three numbers is zero if all are zero individually.

$\Rightarrow$ $a=1,b=1,c=1$

∴ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$

Example: Find the number of all possible matrices of order $3\times 3$with each entry 0 or 1 .

Answer:

Given matrix of order $3\times 3$

∴ Total number of entries = 9

Each entry has two choices either  0 or 1 

Thus the total possible matrices $=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=512$

FAQs of Matrix

Question 1. How do we find the total number of elements in a matrix?

Answer: The total number of elements of any matrix is the product of the number of rows and columns.

Question 2.What do you mean by vector matrix?

Answer: A vector is a one-row or one-column matrix.

Question 3. Is a matrix a scalar or a vector?

Answer: Matrixes are essentially vectors that have been expressed in a two-dimensional table format.

Question 4. Does Trace exist for a rectangular matrix?

Answer: No. Traces exist only for square matrices.

NCERT Class 12 Maths Chapters