The order of a matrix represents its dimensions mathematically. By inspecting the order of matrices, it is much easier to know the number of elements and classify the matrixes. Knowing the order of a matrix is an essential foundation when studying the matrix.
This is why this article focuses on this core component of matrices. Learn how to find the order of a given matrix, classify matrices using their dimensions, and use them to solve more complex problems.
What Is the Order of a Matrix?
The order of a matrix is a system of indicating the number of columns and rows present in a given matrix using algebraic expression. It highlights the order of the matrix’s elements and showcases the matrix’s dimension. In general, the order of a matrix is written as a product of two factors: the first factor represents the number of rows and the second factor represents the number of columns.
Before diving deeper into this topic, recall that a matrix is an array of numbers arranged in a given number of rows and columns. The numbers found inside a matrix are called the elements of the matrix. Now that we’ve reviewed what we know of matrices, take a look at the matrix, A, that has i rows and j columns. This matrix has an order of “i by j” or algebraically, i \times j.
When seeing the matrix notation, A_{i\times j}, what it tells us is that A is a matrix defined by the order, i\times j. Not only does an order of a matrix indicate the number of rows and columns, but it also tells the number of elements present in the matrix: ij.
The order of a matrix is a systematic approach to describing and classifying matrices. This is why it’s helpful to learn about the common matrices that are described by their order.
Types of Matrices Defined by the Order of Matrix
There are different types of matrices that are classified based on their order such as the row and column matrices. Here are some common types of matrices that you’ll encounter in linear algebra:
Row Matrix: This matrix only has one row, so its order is defined as 1 \times j, where j represents the number of columns present in the row matrix.
A_{1\times j} = \begin{bmatrix} a_{11}&a_{12}&a_{13}&.......&a_{1j}\end{bmatrix}For example, the matrix, M_{1\times 4} = \begin{bmatrix} -4& 2& 4& 2\end{bmatrix}, is a row matrix with 1 row and 4 columns. This also means that M is a row matrix with 4 elements.
Column Matrix: Similarly, the column matrix only has one column. The matrix’s order is defined as i \times 1, where i represents the number of rows present in the column matrix.
A_{i\times 1} = \begin{bmatrix} a_{11}\\a_{21}\\a_{31}\\.\\.\\.\\.\\a_{i1}\end{bmatrix}One example of a column matrix is M_{3\times 1} = \begin{bmatrix} 1\\0\\-1\end{bmatrix}. The order of the matrix shows that the matrix has 3 rows and 1 column.
Square Matrix: From its name, the square matrix exhibits the same number of rows and columns. This means that the order of the square matrix will be j \times j. Meaning, it has j rows and j columns.
A_{3 \times 3} = \begin{bmatrix}a_{11} &a_{12} &a_{13}\\ a_{21} &a_{22} &a_{23}\\ a_{31} &a_{32} & a_{33}\end{bmatrix}Take a look at the matrix shown above. This matrix, A, is a square matrix with an order of 3\times 3. Meaning, it has 3 rows and 3 columns as well as 9 elements in total. Another example of a square matrix is M, where M_{2 \times 2} = \begin{bmatrix}1 &-2\\ -4 & 2 \end{bmatrix} .
Rectangular Matrix: Unlike a square matrix, the rectangular matrix has an unequal number of rows and columns. In general, the rectangular matrix will have an order of i\times j. This means that a rectangular matrix will have i rows and j columns, where i and j are unique.
A_{4 \times 3} = \begin{bmatrix}a_{11} &a_{12} &a_{13}\\ a_{21} &a_{22} &a_{23}\\ a_{31} &a_{32} & a_{33}\\ a_{41} &a_{42} & a_{43}\end{bmatrix}Take a look at the general form of a rectangular matrix with four rows and three columns. The order of this matrix is 4\times 3. Another example of a rectangular matrix is M_{2 \times 1} = \begin{bmatrix}-2 &1 &2 &-4\\ -3 &2 &0 &-1\end{bmatrix}, where the order is 2\times 1.
There are other ways to classify matrices, but these are simply four types of matrices easily defined by their orders. Wrapping up this discussion, use what you’ve learned to classify different matrices.
Problem 1
Classify the following matrices as a row matrix, column matrix, square matrix, or rectangular matrix.
a. A= \begin{bmatrix}2 &-1 &0 &-4\\ 4 -&2 &0 &-3\\-3 &-1 &2 &-2\end{bmatrix}
By inspecting the matrix, A, has three rows and four columns. This means that the numbers of rows and columns are not the same, so the matrix is a rectangular matrix.
b. B= \begin{bmatrix}4 &-2 &5 \end{bmatrix}
The matrix, B, has only one row and three columns. Recall that row matrices are matrices with only one row, so B is a row matrix.
c.C= \begin{bmatrix}2 &-1 \\0 &-2 \end{bmatrix}
Now taking a look at the matrix, C, it shows two rows and two columns. When a matrix shares the same number of rows and columns, it can be classified as a square matrix. This means that C is a square matrix.
After understanding the order of a matrix and knowing how to classify common types of matrices, it’s time to learn how to easily identify the order of a given matrix.
How To Find the Order of a Matrix?
Find the order of the matrix by counting its number of rows then counting the number of columns. Write down its order by writing it as i\times j, where i represents the number of rows and j represents the number of columns. This means that when given the matrix, M_{ij}, follow the steps below to find its order:
- Count the number of rows present in the matrix, M, and assign that number as i or the first factor of the order.
- Now count the number of columns present in M and let that be j. This is the second factor for the matrix’s order.
- Write the order of M by multiplying the two numbers, i and j. Hence, the order of the matrix is i \times j
The total number of elements present in a given matrix equals the product of the two factors from the matrix’s order. It is also possible to determine the order of the resulting matrix from performing arithmetic operations on two or more matrices.
Order of Matrix During Addition or Subtraction: Two matrices can also be added or subtracted if their orders are the same. This means that the matrices must share the same number of rows and columns.
| Matrices That Can Be Added or Subtracted | Matrices That Can’t Be Added or Subtracted |
| \begin{aligned}\begin{bmatrix}2 &-1 \\0 &-2 \end{bmatrix} &+\begin{bmatrix}4 &-2 \\1 &0 \end{bmatrix} \\\\\begin{bmatrix}5 &-4 \\6 &-2\\1 &-4 \end{bmatrix} &-\begin{bmatrix}2 &0 \\1 &4\\4 &0 \end{bmatrix}\end{aligned} | \begin{aligned}\begin{bmatrix}4 &-2 \\0 &-3 \end{bmatrix} &+\begin{bmatrix}4 &-2 \\1 &0 \\0 &-4 \end{bmatrix} \\\\\begin{bmatrix}2 &-3 \\4 &-2\\1 &-2 \end{bmatrix} &-\begin{bmatrix}2 &0 \\1 &4 \end{bmatrix}\end{aligned} |
Since matrices have this rule for addition and subtraction, the order of the resulting matrix will be the same with the two matrices.
Order of Matrix During Multiplication: Two matrices can only be multiplied if the first matrix’s column and the second matrix’s row share the same number of elements. The order of the resulting matrix is will reflect the number of rows present in the first matrix and the second matrix’s second column.
\begin{aligned}A_{i\times j} \cdot B_{j\times n} &= AB_{i \times n}\end{aligned}This that when a matrix with an order of 2\times 4 is multiplied to a matrix with an order of 4 \times 5 , the resulting product is a matrix with an order of 2\times 5.
Now that the rules for finding the order of a matrix have been established, it’s time to try out different problems that involve matrices and their order.
Problem 2
What is the order of the matrix, C= \begin{bmatrix}2 &-2 &0 &-3\\ 3& -2 &0 &-3\\-3 &-1 &2 &-2\\-1 &-3 &3 &-2\end{bmatrix} ? Use the order to find the number of elements present in C.
First, count the number of rows present in the matrix, C. The row of a matrix represents the horizontal arrays of numbers present, so C has four rows. Now, recall that the column of the matrix are the vertical array of numbers present, so C has four columns as well.
C= \begin{bmatrix}2 &-2 &0 &-3\\ 3& -2 &0 &-3\\-3 &-1 &2 &-2\\-1 &-3 &3 &-2\end{bmatrix}In fact, C is a square matrix with four rows and columns each. Use this to write the order of C: 4 \times 4. To find the number of elements in C, multiply the order’s components. This means that C has 4 \times 4 = 16 elements in total.
Problem 3
When a matrix has 12 elements, write down all the possible orders of the matrix.
Recall that number of elements of a matrix is simply the product of the matrix’s order. This means that if the matrix has 12 elements, the number of rows and columns present in the matrix will have a product of 12. Write down the factors of 12 to find all the possible orders of the matrix.
Let i be the number of rows and j be the number of columns. The table below summarizes the possible orders of the matrix:
| Rows and Columns | Order | Rows and Columns | Order |
| i = 1, j = 12 | 1 x 12 | i = 6, j = 2 | 6 x 2 |
| i = 1, j = 12 | 12 x 1 | i = 3, j = 4 | 3 x 4 |
| i = 2, j = 6 | 2 x 6 | i = 4, j = 3 | 4 x 3 |
This means that there are 6 possible orders for a matrix with 12 elements.