Elements of Matrix

Elements of the Matrix are the components or entries of the Matrix. These elements could be any variables, numbers, a mix of variables and numbers, or any other kind of special characters as well. The number of elements of a matrix is equal to the product of the number of rows and columns present in a matrix.

This article explores the concept of the element of matrix in detail and makes it easy to grasp for all the readers of the article without any regard to their academic level. All subtopics such as their meaning, definition, symbol, example, and many many more, are covered in the article with plenty of examples. So, let's start our journey to the land of elements of the matrix and understand this concept.

What is Matrix?

Before going in depth details for matrix elements let us first know about the matrix. A matrix is an array of numbers that has been set up in rows and columns to make a rectangular shape. In addition to numerous mathematical disciplines, matrices find extensive use in the fields of engineering, physics, economics, and statistics.

In the past, it was not the matrix that was originally seen, but rather a specific number connected to a square array of numbers known as the determinant.

What are Elements of a Matrix?

The matrix's entries are its only constituent parts or elements. They can be any type of character found inside the matrix, including numbers, variables, mathematical expressions, and more.

In order to better understand elements of Matrix let us take an example. Let us suppose a Matrix A =

\begin{bmatrix} 1 & 2 \\ -3 & 4 \end{bmatrix}

Here elements of matrix A are 1, 2, -3 and 4.

Check: Matrix Addition

Examples of Element of Matrix

As we already know, that every entity in a matrix is called element of the matrix. Let's consider some examples for better understanding.

Example 1: What are elements of matrix, B = \bold{\begin{bmatrix} b & c \\ 3 & 4x \end{bmatrix}} ?

Answer:

Here the elements of the given matrix B are all the elements present inside matrix B that is b, c, 3 and 4x.

For a 2 × 2 matrix, \begin{bmatrix} x & x+y \\ 2 & -4 \end{bmatrix} .

Its elements are are x, x + y, 2 and -4.

For a 3x3 matrix, \begin{bmatrix} x & x+y & -1 \\ 1 & 4 & 3\\ 2 & -4 & 6 \end{bmatrix} .

Its elements are x, x+y, -1, 1, 4, 3, 2, -4 and 6.

Note: While writing the elements of any matrix, the order of the elements are not important.

Types of Elements of Matrix

The components of the matrix are what are known as the matrix's elements and they could be variables, numbers, a mix of the two, or any other kind of special character. Based on the position of these elements in the matrix, we can categorise these elements as:

• Diagonal Elements
• Off-Diagonal Elements

Let's discuss these types in detail.

Diagonal Elements

A diagonal matrix is a square matrix in which all members are zero except for the primary diagonal elements. When dij = 0 and i is not equal to j, a square matrix D = [dij]n x n is referred to as a diagonal matrix.

For example if A =

\begin{bmatrix} a \ 0 & \\ 0\ b & \end{bmatrix}

In the above 2x2 matrix, matrix A is a diagonal matrix. Here 'a' and 'b' are the diagonal elements.

Off Diagonal Elements

The fact that the matrix's row number and column number are not equal represents the off-diagonal members. The first element of the first row and the last element of the last row combine to generate a major diagonal. The other elements in the matrix are referred to as non-diagonal or off-diagonal elements whereas the elements that are on the leading diagonal are referred to as diagonal elements.

For Example: If A =

\begin{bmatrix} a \ 1 & \\ 0\ b & \end{bmatrix} 

In the above 2x2 matrix, matrix A is a diagonal matrix. Here '1' and '0' are the off diagonal elements.

For Example: If B =

\begin{bmatrix} a \ 1\ 0 & \\ c\ d\ 3 &\\ 0\ -1\ b & \end{bmatrix}

In the above 3x3 matrix, matrix B is a diagonal matrix with 1, 0, c, 3, 0 and -1 as off-diagonal elements.

Check: Types of Matrices

Number of Elements of a Matrix

Let us understand the number of elements of a matrix concept by considering a simple example. Let a Matrix C =

\begin{bmatrix} 1 \ 2 & \\ 3\ 4 & \end{bmatrix}

Here matrix C has 2 rows and 2 columns. So, the number of elements of this matrix is 2 × 2 = 4

So, in order to find the number of elements of a matrix we just need to find the product of a number of rows with a number of columns present in the given matrix. A matrix's element count is independent of the kind of items it contains.

For Example, The number of elements in a matrix for the following are:

• For 2 by 2 Matrix, [2 rows and 2 columns]
  •  Thus, the number of elements = 2 × 2 = 4 elements.
• For 2 by 2 Matrix, [2 rows and 3 columns]
  • Thus, the number of elements = 2 × 3 = 6 elements.
• For 3 by 3 Matrix, [3 rows and 3 columns]
  • Thus, the number of elements = 3 × 3 = 9 elements.

Example: Find the number of elements of a matrix with 6 rows and 4 columns.

Answer:

We know,

Number of Elements of a Matrix = Number of rows x number of columns

Number of Elements = 6 × 4 = 24 elements

It is important to note that if a matrix A has m rows and n columns, then the order of the matrix is expressed as m x n, and the number of elements is equal to the product of m and n (i.e., m n).

Check: Trace of a Matrix

Positions of Elements of Matrix

Each component of a matrix has a specific location that is identified by its row number, followed by its column number, sometimes separated by a comma, in the subscript of the alphabet that serves as the matrix's representation. We can also say that the element of a matrix A is present in the i^th row and j^th column which can be shown by A_ij^.

For example if A =

\begin{bmatrix} 1 \ 0 & \\ 3\ 4 & \end{bmatrix}

then:

1 is the element in the 1st row and 1st column. It is written as A₁₁ element of A.

0 is the element in the 1st row and 2nd column. It is written as A₁₂ element of A.

3 is the element in the 2nd row and 1st column. It is written as A₂₁ element of A.

4 is the element in the 2nd row and 2nd column. It is written as A₂₂ element of A.

It is important to note that we need to write the row number first and then the column number and do not write in the reverse order.

Example: Find the position of element 4 and 5 in the given matrix A =

\begin{bmatrix} 2 & 3& \\ 4 & 0& \\ 6 & 5& \end{bmatrix}

Solution:

4 is the element in the 2nd row and 1st column. It is written as A₂₁ element of A .

5 is the element in the 3rd row and 2nd column. It is written as A₃₂ element of A.

Check: Elements of Matrix

Elements in Equal Matrices

For any two given matrices which are equal their corresponding elements are also equal.

For Example: Let A =

\begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}

and Let B =

\begin{bmatrix} x & y \\ z & w \end{bmatrix}

So, here as the corresponding elements are equal so x=1, y=2, z=3, w=1.

Properties of Matrix Elements

The following are the properties of matrix elements as given below:

• The number of elements of a matrix is given by order of m × n.
• Position of an element of a given matrix let say A is represented by A with row number , column number in order as a subscript.
• If two matrices are given as equal then their corresponding elements are also equal.
• For a square matrix the number of elements is always a perfect square.
• Similarly it is obvious that for a rectangular matrix the number of elements is never a perfect square.

Solved Problem on Elements of Matrix

Problem 1: Find all the elements of matrix A \bold{\begin{bmatrix} -2 & 3 \\ 4 & 0 \\ 6 & -5 \end{bmatrix}}  .

Solution:

Elements of the given matrix A are -2, 3, 4, 0, 6 and -5.

Problem 2: Find the position of 4 and -5 in the matrix A \bold{\begin{bmatrix} -2 & 3 \\ 4 & 0 \\ 6 & -5 \end{bmatrix}}  .

Solution:

4 is present in 2^nd row and 1^st column so its position is A₂₁.

-5 is present in 3^rd row and 2^nd column so its position is A₃₂.

Problem 3: For any two given matrices \bold{A = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}}  and Let B = \bold{B = \begin{bmatrix} x & 2 \\ z & 1 \end{bmatrix}}  . If A = B, then find the value of x and z.

Solution:

For any two given matrices which are equal their corresponding elements are also equal.

Thus, for A = B

\Rightarrow \bold{ \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} x & 2 \\ z & 1 \end{bmatrix}}

Thus, x = 1, z = 3

Problem 4: For any two given matrices \bold{A = \begin{bmatrix} 1 & 0 & 0\\ 3 & 1 & 2 \\ -1 & 1 & 4 \end{bmatrix}}  and Let B = \bold{B = \begin{bmatrix} x & 2 & 1\\ z & 1 & 0 \\ y & 3 & 8 \\ \end{bmatrix}}  . If A = B, then find the value of x and z.

Solution:

For any two given matrices which are equal their corresponding elements are also equal.

Thus, for A = B

x= 1, y =-1, z= 3

Also Check:

Practice Problems on Elements of Matrix

Problem 1: For any matrix X = \begin{bmatrix} 2 & 4 & 1\\ 0& 3& -2 \\ 5&1&7\end{bmatrix} . What is the element in the second row and third column?

Problem 2: Given a square matrix A = \begin{bmatrix} 4 & 2 & 1\\ 0& 5& 2 \\ 3&0&6\end{bmatrix} . Find the diagonal and off-diagonal elements of A.

Problem 3: For a matrix  C = \begin{bmatrix} 2 & 4 & 1\\ 0& 3& -2 \\ 2&1&7\end{bmatrix} . Is the sum of elements in the second row is same as the sum of elements in the second column?