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Rectangular Matrix

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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 a rectangular matrix.

Let us learn more about the rectangular matrix along with definitions, examples, properties, and operations on it.

What is a 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 rectangle shape. The matrix given below is a rectangular matrix of order “m × n” that has “m” rows and “n” columns. We know that a rectangle is a geometric shape whose length and breadth are different. Similarly, a matrix that has a different number of rows and columns is rectangular, i.e., the number of rows of the matrix is not equal to the number of columns. A matrix “Am×n = [aij]” is said to be a vertical matrix if m>n, i.e., the number of rows is greater than the number of columns. A matrix “Am×n = [aij]” is said to be a horizontal matrix if m<n, i.e., the number of columns is greater than the number of rows. 

Rectangular Matrix

 

Examples of a Rectangular Matrix

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

  • Following matrix is a rectangular matrix of order “3 × 4,” i.e., the given matrix has three rows and four columns.

A_{3\times4} = \left[\begin{array}{cccc} 0 & 1 & 2 & 3\\ 11 & 12 & 13 & 14\\ -1 & -2 & -3 & -4 \end{array}\right]

  • Following matrix is a rectangular matrix of order “2 × 3,” i.e., the given matrix has three rows and four columns.

B_{2\times3} = \left[\begin{array}{ccc} a & b & c\\ p & q & r \end{array}\right]

Types of Rectangular Matrices

There are two types of Rectangular Matrices which are:

  • Row Matrix
  • Column Matrix

Let’s learn about them in detail.

Row Matrix

A row matrix is defined as a matrix that has only one row. A matrix “A = [aij]” 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.”

P_{1\times4} = \left[\begin{array}{cccc} a & b & c & d\end{array}\right]

Column matrix

A column matrix is defined as a matrix that has only one column. The matrix “A = [aij]” 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.”

Q_{3\times1} = \left[\begin{array}{c} p\\ q\\ r \end{array}\right]

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 ExampleM = \left[\begin{array}{ccc} 1 & 2 & 3\\ 0 & 4 & 5 \end{array}\right]_{2\times3} + N = \left[\begin{array}{cc} -9 & -7\end{array}\right]_{1\times2}   is not possible as the order of two matrices is different.

P = \left[\begin{array}{c} 17\\ 23 \end{array}\right]_{2\times1} - Q = \left[\begin{array}{c} -8\\ 5 \end{array}\right]   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, \left[\begin{array}{cc} 3 & 4\end{array}\right]_{1\times2} \times\left[\begin{array}{c} -1\\ 5 \end{array}\right]_{2\times1}   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 AT. 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.

A = \left[\begin{array}{cccc} a & b & c & d\\ p & q & r & s \end{array}\right]_{2\times4} ⇒ A^{T} = \left[\begin{array}{cc} a & p\\ b & q\\ c & r\\ d & s \end{array}\right]_{4\times2}

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.
  • A rectangular matrix will not have an adjoint.
  • As the adjoint and determinant are defined for a rectangular matrix, it will not have an inverse.
  • 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.

Read More,

Solved Examples on Rectangular Matrix

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

A_{3\times4} = \left[\begin{array}{cccc} 0 & -1 & 3 & 5\\ 7 & 9 & -4 & 2\\ 11 & 8 & 0 & 13 \end{array}\right]

Solution:

The given matrix has 3 rows and 3 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.”

A^{T} = \left[\begin{array}{ccc} 0 & 7 & 11\\ -1 & 9 & 8\\ 3 & -4 & 0\\ 5 & 2 & 13 \end{array}\right]_{4\times3}

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

P = \left[\begin{array}{cccc} 3 & 5 & 7 & 9\\ 2 & 4 & 6 & 8 \end{array}\right] and Q = \left[\begin{array}{cccc} -5 & 0 & 3 & 8\\ 14 & 9 & -7 & 6 \end{array}\right]

Solution:

P + Q = \left[\begin{array}{cccc} 3 & 5 & 7 & 9\\ 2 & 4 & 6 & 8 \end{array}\right]_{2\times4} + \left[\begin{array}{cccc} -5 & 0 & 3 & 8\\ 14 & 9 & -7 & 6 \end{array}\right]_{2\times4}

P + Q = \left[\begin{array}{cccc} (3-5) & (5+0) & (7+3) & (9+8)\\ (2+14) & (4+9) & (6-7) & (8+6) \end{array}\right]_{2\times4}

P + Q = \left[\begin{array}{cccc} -2 & 5 & 10 & 17\\ 16 & 13 & -1 & 14 \end{array}\right]_{2\times4}

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

  • A = \left[\begin{array}{ccc} 3 & 0 & -6\\ 5 & 9 & 7 \end{array}\right]_{2\times3}
  • B = \left[\begin{array}{ccc} 0 & 1 & 2\\ 4 & 5 & 6\\ 7 & 8 & 9 \end{array}\right]_{3\times3}

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

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

M = \left[\begin{array}{cc} 1 & 5\\ 2 & 6\\ 3 & 8 \end{array}\right]_{3\times2} and N = \left[\begin{array}{ccc} 5 & 0 & 6\\ 4 & -3 & 1 \end{array}\right]_{2\times3}

Solution:

M × N = \left[\begin{array}{cc} 1 & 5\\ 2 & 6\\ 3 & 8 \end{array}\right]_{3\times2} \times\left[\begin{array}{ccc} 5 & 0 & 6\\ 4 & -3 & 1 \end{array}\right]_{2\times3}

M × N = \left[\begin{array}{ccc} (5+20) & (0-15) & (6+5)\\ (10+24) & (0-18) & (12+6)\\ (15+32) & (0-24) & (18+8) \end{array}\right]_{3\times3}

M × N = \left[\begin{array}{ccc} 25 & -15 & 11\\ 34 & -18 & 18\\ 47 & -24 & 26 \end{array}\right]_{3\times3}

FAQs on Rectangular Matrix

Question 1: Define a rectangular matrix.

Answer:

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 rectangle shape.

Question 2: What is a row matrix?

Answer:

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

Question 3: What is a column matrix?

Answer:

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

Question 4: What is meant by the transpose of a matrix?

Answer:

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 AT.


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Last Updated : 26 Mar, 2023
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