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Order of Matrix

The order of matrix represents the number of rows and columns in the matrix. A matrix is an array of elements arranged in rows and columns, and the order of matrix helps in getting a count of the rows and columns in a matrix. Further, the order of matrix helps in knowing the type of the matrix, and the total number of elements in a matrix.

The order of a matrix is an important aspect that helps to decide if a particular arithmetic operation can be performed across two matrices. Here, based on the order of matrix, we can learn about the different types of matrices, and the different arithmetic operations which can be performed across matrices.

1.	What is Order of Matrix?
2.	Type of Matrices Based on Order of Matrix
3.	Order of Matrix for Different Matrix Operations
4.	Examples on Order of Matrix
5.	Practice Questions
6.	FAQs on Order of Matrix

What is Order of Matrix?

The order of matrix gives the dimension of the matrix, and it informs the number of rows and columns present in the matrix. The order of matrix is general represented as A_{m × n}, where m is the number of rows, and n is the number of columns in the given matrix. Also, the multiplication answer of the order of matrix (m × n) gives the number of elements in the matrix.

Order of the Matrix

In the above matrix, we can observe m number of rows and n number of columns. The first number in the order of matrix always represents the number of rows, and the second number in the order of matrix always represents the number of columns in the matrix.

Type of Matrices Based on Order of Matrix

The order of matrix gives the dimensions of the matrix, and it defines the different types of matrices. Let us check the order of some of the different types of matrices.

Order of a Row Matrix: A row matrix has one row and numerous columns. Hence the order of row matrix is of the form 1 × n. \[A_{1 × n} = \begin{bmatrix}a_1&a_2&a_3&\cdots&a_n\end{bmatrix}\]
Order of a Column Matrix: A column matrix has one column and numerous rows. Hence the order of column matrix is n × 1.\[A_{n × 1} = \begin{bmatrix}a_1\\a_2\\a_3\\ \vdots\\a_n\end{bmatrix}\]
Order of a Square Matrix: A square matrix as the name suggests, has an equal number of rows and columns. Hence the order of a square matrix is of the form n × n. Here we have an equal number of 3 rows and 3 columns in the below matrix. \[A_{3×3} = \begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\]
Order of a Rectangular Matrix: A rectangular matrix has an unequal number of rows and columns and hence the order of a rectangular matrix is of the form m × n. Here in the below matrix, we have 2 rows and 3 columns. \[A_{2×3} = \begin{bmatrix}a&b&c\\ \\ d&e&f\end{bmatrix}\]
Order of a Transpose Matrix: The transpose of a matrix is obtained by changing its rows into columns and its columns into rows. For a matrix of order m × n, the order of transpose of the given matrix is n × m. Here the given matrix has 2 rows and 3 columns, and the transpose of this matrix has 3 rows and 2 columns. \[A_{2×3} = \begin{bmatrix}a&b&c\\d&e&f\end{bmatrix}, A^T_{3×2} = \begin{bmatrix}a&d\\b&e\\c&f\end{bmatrix}\]

Order of Matrix for Different Matrix Operations

The order of matrix refers to the type of the matrix. Further many of the arithmetic operations of matrices are based on the order of the referred matrix. Let us check how the following operations on matrices are performed based on the order of the .matrix.

Addition and Subtraction of Matrices: For the addition or subtraction of two matrices, the order of both the matrices should be equal. The number of rows in both the matrices should be equal, and the number of columns in both the matrices should also be equal. Let us understand this with a simple example.

\( \left[\begin{array}{ccc}
2 & {-1} & 3 \\ \\
0 & 5 & 2 \end{array}\right]_{\!2 \!× \!3} + \left[\begin{array}{ccc}
\!0 &\! 2 & \!7\\ \\
\!1 & \!-2 & \!9
\end{array}\right]_{\!2 \!× \!3}\)

\(= \left[\begin{array}{ccc}
2+0 & {-1} \!+\!2 & 3 \!+\!7\\ \\
0\!+\!1 & 5\!+\!(-2) & 2\!+\!9
\!\end{array}\right] \)

\(= \left[\begin{array}{ccc}
\!2 & \!1 & \!10 \\ \\
\!1 & \!3 & \!11
\end{array}\right]_{\!2 \!× \!3} \)

Here the corresponding elements of both the matrices are added to obtain the elements of the resultant matrix, and hence the number of elements and the order of both the matrices should be equal. The order of both the matrics in the above addition of matrices is 2 × 3.

Multiplication of Matrices: The multiplication of matrices involves a special condition of the order of matrix. The number of columns in the first matrix for multiplication should be equal to the number of rows in the second matrix. Further, the order of the resultant matrix is equal to the number of rows of the first matrix and the number of columns of the second matrix.

\(AB=\left[ \begin{matrix} \!1 & \!2 & \!3 \\ \\ \!3 & \!1 & \!0 \\\end{matrix} \right]_{\!2 \!× \!3} \times \left[ \begin{matrix} \!1 & \!4 \\ \!3 & \!-1 \\ 2 & -3 \\\end{matrix} \right]_{\!3 \!× \!2}\)

\(=\begin{bmatrix}\!1\!×\!1 \!+ \!2\!×\!3 \!+ \!3\!×\!2 &\!1 \!×\!4 \!+\! 2\!×\!(-1) \!+\!3\!×\!(-3)\\\!3\!×\!1 \!+ \!1\!×\!3 \!+ \!0\!×\!2 &\!3 \!×\!4 \!+ \!1\!×\!(-1) \!+\!0\!×\!(-3)\end{bmatrix}\)

\(=\left[ \begin{matrix} \!13 & \!-7\\ \\ \!6 & \!11 \\ \end{matrix} \right]_{\!2 × \!2}\)

In the above example, the number of columns in the first matrix and the number of rows in the second matrix is equal to 3. And the order of the resultant matrix is 2 × 2 as it has 2 rows (which is equal to the number of rows in the first matrix) and 2 columns (which is equal to the number of columns in the second matrix).

☛ Related Topics:

The following related topics are helpful for a better understanding of this concept of order of matrix.

Important Notes on Order of Matrix:

The following points summarize some of the key points learned about the order of matrix.

In the order of matrix m × n, the first number m always represents the number of rows, and the second number n always represents the number of columns.
For the addition and subtraction of two matrices, the order of matrices should be equal.
For the multiplication of two matrices, the number of columns in the first matrix should be equal to the number of rows in the first matrix.
In the multiplication of two matrices, the order of the resultant matrix is such that the number of rows is equal to the first matrix, and the number of columns is equal to the second matrix.

Examples on Order of Matrix

Example 1: The order of a matrix is 4 x 3. What is the order of a matrix with which it can be multiplied? Also, what is the order of the resultant matrix after multiplication?

Solution:

The order of the given matrix is 4 x 3. Let us represent this matrix as \(A_{4 × 3}\). Also, let us consider another matrix B with which this matrix is to be multiplied.

The condition for matrix multiplication is that the number of columns in the first matrix should be equal to the number of rows in the second matrix. Since we have 3 columns in matrix A, there would be 3 rows in matrix B. Let us now take the order of matrix B as 3 x 2.

\(A_{4 × 3} \times B_{3 × 2} = C_{4 × 2} \)

As a result of matrix multiplication, the resultant matrix C will have the number of rows of the first matrix and the number of columns of the second matrix.

Answer: Therefore, the order of the second matrix can be 3 x 2, and in this case, the order of the resultant matrix is 4 × 2.
Example 2: Find the order of matrix obtained on multiplying two matrices having the order of 2 × 4, and 4 × 3, respectively.

Solution:

Let us represent the order of the given two matrices as \(A_{2 × 4}\), and \(B_{4 × 3}\) respectively.

Here it satisfies the first condition of multiplication of matrices, where the number of columns in the first matrix is equal to the number of rows in the second matrix and is equal to 4.

Further, the order of the resultant matrix is equal to the number of rows in the first matrix and the number of columns in the second matrix. Hence the order of the resultant matrix is 2 × 3.

\(A_{2 × 4} \times B_{4 × 3} = C_{2 × 4}\)

Answer: Hence the order of the resultant Matrix is 2 × 3.
Example 3: If A is a matrix of order 3 × 4 and B is a matrix of order 4 × 3, is it possible to add them?

Solution:

The condition for two matrices to be added is that their orders should be the same.

But here, the order of A is not equal to that of B.

So A + B is not possible.

Answer: Finding the sum of matrices is not possible.

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Practice Questions on Order of Matrix

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FAQs on Order of Matrix

How to Find the Order of Matrix?

The order of matrix can be easily calculated by checking the arrangement of the elements of the matrix. A matrix is an arrangement of elements arranged as rows and columns. The order of matrix is written as m × n, where m is the number of rows in the matrix and n is the number of columns in the matrix.

How Can We Find the Number of Elements From the Order of Matrix?

The order of matrix is written as m × n, and this product gives the number of elements in the matrix. As an example let us consider a matrix of order 2 × 3, and this product 2 × 3 = 6 is the number of elements in the matrix.

What is the Condition for Order of Matrix for Addition and Subtraction?

To add/subtract two or more matrices, they should be of the same order. For example, a matrix of order 2 x 3 cannot be added to a matrix of order 3 x 2.

What is the Condition for Order of Matrix for Matrix Multiplication?

The important condition for multiplication based on the order of matrix is that the number of columns in the first matrix is equal to the number of rows in the second matrix. Also, the resultant matrix of multiplication of matrices has the number of rows equal to the number of rows in the first matrix, and the number of columns equal to the columns in the second matrix.

How Do You Find the Types of Matrix Based on Order of Matrix?

The order of a matrix is represented as m × n, and here m represents the number of rows and n represents the number of columns in the given matrix. Based on the values of m and n the size of the matrix varies. The different matrices which can be identified based on the order of the matrix is as follows.

Row Matrix. Here the value of m = 1, and it has one row and more than one column. The order of the row matrix is 1 × n.
Column Matrix: Here the value of n = 1, and the matrix has one column and more than one row. The order of the column matrix is m × 1
Square Matrix: Here the matrix has an equal number of rows and columns. m = n. The order of the square matrix is n × n.
Rectangular Matrix: In this matrix, the number of rows is different from the number of columns. m ≠ n. The order of the

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