Elements of Matrix

The elements of matrix are nothing but the components of matrix. They can be numbers, variables, a combination of both, or any special characters. The number of elements of matrix is equal to the product of number of rows and number of columns in it.

Let us learn more about elements of matrix along with more examples.

The elements of matrix are nothing but the entries of the matrix. They can be numbers, variables, any mathematical expressions, or any other characters inside the matrix. For example, the elements of a matrix A = \(\left[\begin{array}{rl}
5 & 2 \\
x &y \\
-1 & 3
\end{array}\right]\) are 5, 2, x, y, -1, and 3. Here are more examples:

• The elements of B = \(\left[\begin{array}{rl}
  a &1 \\ \\
  -1 & 5
  \end{array}\right]\) are a, 1, -1, and 5.
• The elements of C = \(\left[\begin{array}{rl}
  1 & 2 & 5x + 3\\
  3 &-2 & 2 \\
  1 & 3 & x
  \end{array}\right]\) are 1, 2, 5x+3, 3, -2, 2, 1, 3, and x.

While listing the elements of a matrix as shown in the examples the order doesn't matter.

In the above examples, A has 2 rows 3 columns and it has 6 elements; B has 2 rows and 2 columns and it has 4 elements; and C has 3 rows and 3 columns and it has 9 elements. Observing these examples, did you get an idea of how to calculate the number of elements of a matrix without listing and counting the elements? Yes, it is nothing but the product of the number of rows and number of columns. The number of elements of a matrix is irrespective of the type of elements present in it. For example, the number of elements of a matrix with

• 5 rows and 2 columns is 5 × 2 = 10.
• 5 rows and 2 columns is 2 × 5 = 10.
• 3 rows and 4 columns is 3 × 4 = 12.
• 4 rows and 3 columns is 4 × 3 = 12.

If the number of rows and the number of columns of a matrix A are m and n respectively then its order is written as m × n and the number of elements is the product of m and n (i.e., mn).

Every element of a matrix has a unique position and is determined by its row number followed by column number (separated by a comma sometimes) in the subscript of the alphabet that represents the matrix. i.e., the element of a matrix A that is present in i^th row and j^th column is represented as A ᵢⱼ (or) Aᵢ,ⱼ (or) (i, j)^th element of A. For example if A = \(\left[\begin{array}{rl}
5 & 2 \\
x &y \\
-1 & 3
\end{array}\right]\) then:

• 5 is the element in the 1^st row and 1^st column. It is written as A₁₁ (or) (or) A₁,₁ (or) (1, 1)^th element of A .
• 2 is the element in the 1^st row and 2^nd column. It is written as A₁₂ (or) A₁,₂ (or) (1, 2)^th element of A.
• -1 is the element in the 3^rd row and 1^st column. It is written as A₃₁ (or) A₃, ₁ (or) (3, 1)^th element of A.

Remember to write the row number first and then the column number (not in the reverse order) while writing the position.

• The position of an element of a matrix A is represented by A with the subscript (row number, column number).
• If two matrices A and B are equal, then their corresponding elements are equal. i.e., Aᵢ,ⱼ = Bᵢ,ⱼ, for every i, j.
• The number of elements of a matrix of order m x n is mn.
• The number of elements of a square matrix is always a perfect square.
• The number of elements of a rectangular matrix is never a perfect square.

☛ Related Topics:

• Matrix Calculator
• Properties of Matrices
• Types of Matrices
• Matrix Multiplication

How Can We Find the Elements of Matrix?

The elements of matrix are just the components present inside the matrix and are separated by rows and columns. For example, the elements of a matrix \(\left[\begin{array}{rrrr}
1 & 3 \\ \\
0 & 2
\end{array}\right]\) are 1, 3, 0, and 2. Some or all the elements of a matrix can be the same.

How Many Elements a Matrix Has?

The number of elements of a matrix = the number of rows multiplied by the number of columns. For example, if the number of rows is 3 and the number of columns is 4 in a matrix then the number of elements in it is 3 x 4 = 12.

How to Find the Position of a Matrix?

The position of a matrix is the row number followed by the column number where that element is present. For example, in the matrix B = \(\left[\begin{array}{rrrr}
1 & 3 & 1 & -1 \\
0 & 2 & 3 & 2
\end{array}\right]\), 1 is present in the 1^st row and the 3^rd column and hence we can write B₁, ₃ = 1 (or) B₁₃ = 1.

What is the Number of Elements of a Square Matrix?

The number of elements of a square matrix of order n x n is n². Thus, the number of elements of a square matrix is always a perfect square.

How are Elements of Matrix Denoted By?

We represent the elements of a matrix by their positions. Aₘ,ₙ represents the element of matrix A that lies in row m and column n. Here, the row number comes first and then the column number.

What is the Relation Between Elements of Matrix and its Order?

If a matrix has m rows and n columns then its order is written as m x n. If we just multiply m and n then the product gives the number of elements of the matrix.