Rectangular Matrix

A rectangular matrix is defined as a matrix that doesn't have an equal number of rows and columns. It is one of the types of matrices where the arrangement of elements is in a rectangular shape. The matrix given below is a rectangular matrix of order "m × n" that has "m" rows and "n" columns.

Examples of a Rectangular Matrix

Some common examples of rectangular matrices of different orders are given below:

• The following matrix is a rectangular matrix of order "3 × 4," i.e., the given matrix has three rows and four columns.

• The following matrix is a rectangular matrix of order "2 × 3," i.e., the given matrix has two rows and three columns.

Types of Rectangular Matrices

There are two types of Rectangular Matrices which are:

• Row Matrix
• Column Matrix

Row Matrix

A row matrix is defined as a matrix that has only one row. A matrix "A = [a_ij]" is said to be a row matrix if the order of the matrix is "1 × n."

• Matrix given below is a row matrix of order "1 × 4."

Column Matrix

A column matrix is defined as a matrix that has only one column. The matrix "A = [a_ij]" is said to be a column matrix if the order of the matrix is "m × 1."

• Matrix given below is a row matrix of order "3 × 1."

Addition and Subtraction of Rectangular Matrices

Two or more rectangular matrices can be added or subtracted when all the matrices are of the same order.

For Example:

is not possible as the order of two matrices is different.

 is possible as the order of two matrices is the same.

Multiplication of Rectangular Matrices

The multiplication of any two rectangular matrices is possible if and only if the number of columns in the first matrix and the number of rows in the second matrix are equal.

• For example, if A and B are two rectangular matrices, the multiplication of the matrices is possible if the orders of the matrices are "m × n" and "n × p" respectively.
• Then the order of the resultant matrix will be "m × p."
• The product of two rectangular matrices may or may not be rectangular.

For example,  is possible as number of columns in the first matrix and the number of rows in the second matrix is equal.

Transpose of a Rectangular Matrix

The transpose of a matrix is obtained by interchanging its rows into columns or columns into rows.

• If "A" is any matrix of order "m × n," then its transpose is represented either as A' or A^T.
• As the given matrix "A" has "m" rows and "n" columns, its transpose will have "n" rows and "m" columns.
• Have a look at the example given below to understand the transposition of a matrix.

Properties of a Rectangular Matrix

Following are some important properties of a rectangular matrix:

• The order of a rectangular matrix will have two different numbers as it has a different number of rows and columns.
• If the number of rows in a matrix is less than the number of columns, then the matrix is known as a "horizontal matrix."
• If the number of columns in a matrix is less than the number of rows, then the matrix is known as a "vertical matrix."
• Two or more rectangular matrices can be added or subtracted when all the matrices are of the same order.
• The multiplication of any two rectangular matrices is possible if and only if the number of columns in the first matrix and the number of rows in the second matrix is equal.
• The product of two rectangular matrices may or may not be rectangular.
• As the determinant of a matrix is not defined for rectangular matrices, the concepts of singular and non-singular matrices do not apply to rectangular matrices.
• Rectangular matrix is not symmetric, as its transpose matrix will never be equal to the original matrix. For example, if P is a rectangular matrix of order "2 × 3," then the order of its transpose is "3 × 2."
• Rectangular matrix did not have eigenvalues, since eigenvalues are defined only for square matrices.

Related Articles:

• Minors and Cofactors of Determinants
• Determinant of a Matrix
• Adjoint of a Matrix

Solved Questions on Rectangular Matrix

Question 1: Calculate the transpose of the rectangular matrix given below.

Solution:

The given matrix has 3 rows and 4 columns, i.e., its order is "3 × 4." We know that the transpose of a matrix is obtained by interchanging its rows into columns. So, the resultant matrix will have 4 rows and 3 columns, i.e., its order is "4 × 3."

Question 2: Find the sum of the matrices given below.

Solution:

Question 3: Determine whether the matrices given below are rectangular or not.

Solution:

• The given matrix A has 2 rows and 3 columns, i.e., the number of rows is not equal to the number of columns. Hence the given matrix is a rectangular
• The given matrix B has 3 rows and 3 columns, i.e., the number of rows is equal to the number of columns. Hence the given matrix is not a rectangular

Question 4: Find the product of the matrices given below.

Solution:

Unsolved Practice Questions on Rectangular Matrix

Question 1: Check whether the given matrix is rectangular or not. If it is, find its transpose.

Question 2: Find the product of the matrices given below.