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Rectangular Matrix
A rectangular matrix is a matrix that is rectangular in shape. We know that the elements of a matrix are arranged in rows and columns. If the number of rows in a matrix is not equal to the number of columns in it then the matrix is known as the rectangular matrix.
Let us learn more about the rectangular matrix along with definition, examples, properties, and operations on it.
1.  What is Rectangular Matrix? 
2.  Operations on Rectangular Matrix 
3.  Properties of Rectangular Matrix 
4.  FAQs on Rectangular Matrix 
What is Rectangular Matrix?
A rectangular matrix is a matrix in which the number of rows is NOT equal to the number of columns. It is one of the types of matrices. In geometry, a rectangle is a quadrilateral in which the length is different from its width. In the same way, the number of rows of a rectangular matrix is different from the number of columns and hence the name "rectangular". The following is a rectangular matrix of order m x n (i.e., the number of rows = m and the number of columns = n).
Rectangular Matrix Examples
Here are some rectangular matrices and in each matrix, the number of rows is NOT equal to the number of columns.
 \(\left[\begin{array}{llll}
1 & 8 & 3 & 2 \\ \\
5 & 3 & 0 & 6
\end{array}\right]\) has 2 rows and 4 columns is a rectangular matrix of order 2 x 4.  \(\left[\begin{array}{llll}
1 & 7 \\
2 & 4\\
6 & 3 \\
5 & 0
\end{array}\right]\) has 4 rows and 2 columns is a rectangular matrix of order 4 x 2.  \(\left[\begin{array}{llll}
1 & 7 & 8 & 2
\end{array}\right]\) has 1 row and 4 columns is a rectangular matrix of order 1 x 4.
Types of Rectangular Matrix
There can be two types of rectangular matrices: with rows less than columns and with rows more than columns. Here are examples for each of these types:
 Rectangular matrix with rows less than columns: \(\left[\begin{array}{llll}
1 & 8 & 3 & 2 \\ \\
5 & 3 & 0 & 6
\end{array}\right]\)  Rectangular matrix with rows more than columns: \(\left[\begin{array}{llll}
1 & 7 \\
2 & 4\\
6 & 3 \\
5 & 0
\end{array}\right]\)
Operations on Rectangular Matrix
All matrix operations are not always possible on rectangular matrices. Here are few operations that are possible on rectangular matrices with few limitations.
Addition and Subtraction of Rectangular Matrices
The addition and subtraction of two or more rectangular matrices is possible when all of them are of the same order. For example:
 \(\left[\begin{array}{llll}
1 & 8 & 3 \\ \\
5 & 3 & 0
\end{array}\right]_{2 \times 3}\) + \(\left[\begin{array}{llll}
2 & 3 & 5 & 2 \\ \\
1 & 0 & 2 & 6
\end{array}\right]_{2 \times 4}\) is NOT possible.  \(\left[\begin{array}{llll}
1 & 8 & 3 \\ \\
5 & 3 & 0
\end{array}\right]_{2 \times 3}\)  \(\left[\begin{array}{llll}
2 & 3 & 5 \\ \\
1 & 0 & 2
\end{array}\right]_{2 \times 3}\) is possible.
Multiplication of Rectangular Matrices
The multiplication of two rectangular matrices A and B is possible only if the following condition is satisfied: the number of columns of A = the number of rows of B. In other words, A and B should be of the orders m x n and n x p. The product of A and B in this case will be a matrix of order m x p. For example:
 \(\left[\begin{array}{ll}
a & b & c \\
c & d & e \\
\end{array}\right]_{2 \times 3}\) · \(\left[\begin{array}{ll}
x & y
\end{array}\right]_{2 \times 1}\) is NOT possible.  \(\left[\begin{array}{ll}
a & b \\
c & d \\
e & f
\end{array}\right]_{3 \times 2}\) · \(\left[\begin{array}{ll}
x & y
\end{array}\right]_{2 \times 1}\) is possible.
The product of two rectangular matrices may or may not be a rectangular matrix. For example, \(\left[\begin{array}{l}
1 \\
2 \\
6
\end{array}\right]_{3 \times 1}\) · [1 2 3]\(_{1 \times 3}\) is a matrix of order 3 x 3 which is a square matrix.
Transpose of Rectangular Matrix
The transpose of a matrix is obtained by writing its rows as columns (or writing its columns as rows). Hence, if the order of a rectangular matrix is m x n then its transpose is a rectangular matrix of order n x m. For example, the transpose of \(\left[\begin{array}{ll}
a & b & c \\
c & d & e \\
\end{array}\right]_{2 \times 3}\) is \(\left[\begin{array}{ll}
a & c \\
b & d \\
c & e
\end{array}\right]_{3 \times 2}\). So a rectangular matrix is never equal to its transpose and hence is never symmetric.
Properties of Rectangular Matrix
Here are the properties of rectangular matrix based upon its definition.
 In a rectangular matrix, the number of rows and the number of columns is different. Thus, the order of a rectangular matrix is comprised of two different numbers.
 A row matrix or a column matrix with more than one element is always a rectangular matrix.
For example, [1 2 3] is a row matrix of order 1 x 3 and hence it is rectangular.  The determinant of a rectangular matrix is NOT defined. Hence the concepts of singular matrix and nonsingular matrix are NOT applicable for a rectangular matrix.
 A rectangular matrix cannot have an adjoint.
 A rectangular matrix cannot have an inverse as its adjoint and determinant are NOT defined.
 A rectangular matrix cannot be symmetric. Because, for example, the transpose of a rectangular matrix of order 2 x 3 is a matrix of order 3 x 2 and hence they cannot be equal.
 Addition and subtraction of two rectangular matrices is possible only when they are of the same order.
 Multiplication of two rectangular matrices A and B is possible only when the number of columns of A is equal to the number of rows of B.
 The product of two rectangular matrices need not be rectangular.
 A rectangular matrix cannot have eigenvalues (and hence cannot have eigenvectors)
 Identity matrices, diagonal matrices, scalar matrices, orthogonal matrices, symmetric matrices, singular matrices, etc are never rectangular.
Related Topics:
Rectangular Matrix Examples

Example 1: Which of the following are rectangular matrices? (a) \(\left[\begin{array}{llll}
1 & 5 & 8 & 0 \\ \\
2 & 3 & 1 & 6
\end{array}\right]\) (b) \(\left[\begin{array}{llll}
1 & 2 & 3 \\ 4&5&6 \\
5 & 3 & 0
\end{array}\right]\)Solution:
We know that in a rectangular matrix, the number of rows is not equal to the number of columns.
(a) In the given matrix, the number of rows = 2 and the number of columns = 4 and they are not equal.
Hence it is rectangular.
(b) In the given matrix, the number of rows = the number of columns ( = 3).
Hence it is NOT rectangular.
Answer: (a) is rectangular and (b) is NOT rectangular.

Example 2: Find the transpose of the rectangular matrix \(\left[\begin{array}{llll}
1 & 5 & 8 \\ \\
2 & 3 & 1
\end{array}\right]\) and determine whether it is syemmetric.Solution:
The given matrix is, A = \(\left[\begin{array}{llll}
1 & 5 & 8 \\ \\
2 & 3 & 1
\end{array}\right]\).We will write the columns of the given matrix as rows to find its transpose. Then
A^{T }= \(\left[\begin{array}{llll}
1 & 2\\
5 &3\\
8&1
\end{array}\right]\)A is of order 2 x 3 and A^{T} is of order 3 x 2. Clearly, they are NOT equal and hence A is NOT symmetric.
Answer: A^{T }= \(\left[\begin{array}{llll}
1 & 2\\
5 &3\\
8&1
\end{array}\right]\) and A is NOT symmetric. 
Example 3: Which of the following operations of matrices are possible?
(a) \(\left[\begin{array}{llll}
a & b & c \\ \\
d & e & f
\end{array}\right]_{2 \times 3}\) + \(\left[\begin{array}{llll}
1 & 2 & 3 \\ \\
1 & 0 & 2
\end{array}\right]_{2 \times 3}\)
(b) \(\left[\begin{array}{ll}
1 & 3 & 2 \\
0 & 1 & 4 \\
\end{array}\right]_{2 \times 3}\) · \(\left[\begin{array}{ll}
5 & 6
\end{array}\right]_{2 \times 1}\)Solution:
(a) We can add two rectangular matrices when they are of same order.
So the given addition is possible as the order of each matrix is same as 2 x 3.
(b) To multiply two matrices, the number of columns of the first matrix must be same as the number of rows of the second matrix. But here they are not same because
 the number of columns of the first matrix = 3 and
 the number of rows of the second matrix = 2
So the given operation is NOT possible.
Answer: (a) Possible (b) NOT possible.
FAQs on Rectangular Matrix
What is Rectangular Matrix Definition?
A rectangular matrix is a type of matrices in which the number of rows is NOT equal to the number of columns. It is one type of matrices. For example, \(\left[\begin{array}{ll}
2 & 6 \\
3 & 2 \\
8 & 4 \\
0 & 2
\end{array}\right]\) is a rectangular matrix of order 4 x 2.
What are Rectangular Matrix Properties?
Here are the properties of rectangular matrix in brief: A rectangular matrix cannot have determinant, adjoint, and hence inverse. It also cannot have eigenvalues. It cannot be singular or symmetric. For more detailed properties, visit the "Properties of Rectangular Matrix" section of this page.
Is a Square Matrix a Rectangular Matrix?
In geometry, we say that a square is a rectangle but a rectangle need not be a square. In the same way, we can say that a square matrix is rectangular but a rectangular matrix doesn't need to be a square matrix. But in general, the term rectangular matrix is used only when it has an unequal number of rows and columns.
What is the Difference Between a Square and Rectangular Matrix?
The square matrix and rectangular matrix are two different types of matrices.
 In a square matrix, the number of rows is equal to the number of columns.
 In a rectangular matrix, the number of rows is NOT equal to the number of columns.
What are Rectangular Matrix EigenValues?
Eigenvalues can be found only for square matrices. So a rectangular matrix cannot have eigenvalues. Hence, they cannot have eigenvectors.
What is Rectangular Matrix Transpose?
The transpose of a rectangular matrix is obtained by interchanging the rows and columns. The transpose of a rectangular matrix of order p x q is a rectangular matrix of order q x p.
What is Rectangular Matrix Order?
The order of a rectangular matrix is written as "number of rows x number of columns" and is made up of two different numbers. For example, the orders of rectangular matrices can be 2 x 3, 3 x 2, 4 x 5, etc.
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