Column Matrix
Column matrix is a matrix having all its elements in a single column. The elements are arranged in a vertical manner, and the order of a column matrix is n x 1. A column matrix has only one column and can have numerous rows, which are equal to the number of elements in the column.
Let us learn more about the properties of column matrix, the matrix operations on column matrix, through examples, FAQs.
1.  What Is A Column Matrix? 
2.  Properties Of Column Matrix 
3.  Operations On Column Matrix 
4.  Examples on Column Matrix 
5.  Practice Questions 
6.  FAQs on Column Matrix 
What Is A Column Matrix?
Column matrix is a matrix in which all the elements are in a single column. A column matrix has only one column and multiple rows. The order of a column matrix is n × 1, and it has n elements. The elements are arranged in a vertical manner, with the number of elements equal to the number of rows in a column matrix. The general form of a column matrix is as follows.
Examples of Column Matrix
Let us look at the below three examples of column matrices.
B = \(\begin{bmatrix}4\\5\end{bmatrix}_{2×1}\)
C = \(\begin{bmatrix}a\\b\\c\end{bmatrix}_{3×1}\)
D = \(\begin{bmatrix}8\\2\\4\\1\end{bmatrix}_{4×1}\)
Properties of Column Matrix
The following properties of the column matrix, help in a deeper understanding of the column matrix.
 A column matrix has only one column.
 A column matrix has numerous rows.
 The number of elements in a column matrix is equal to the number of rows in the matrix.
 A column matrix is also a rectangular matrix.
 The transpose of a column matrix is a row matrix.
 The column matrix can be added or subtracted to only a column matrix of the same order.
 A column matrix can be multiplied with only a row matrix
 The product of a column matrix with a row matrix gives a singleton matrix.
Operations On Column Matrix
The following algebraic operations of addition, subtraction, multiplication, and division can be performed across column matrices. The addition and subtraction operations on column matrices can be performed as for any other matrices. A column matrix can be added or subtracted to any other column matrix only. Here, the order of the two matrices should be the same.
A = \(\begin{bmatrix}7\\3\\4\\5\end{bmatrix}\), B = \(\begin{bmatrix}3\\8\\2\\7\end{bmatrix}\)
A + B = \(\begin{bmatrix}7+3\\(3)+8\\4+2\\5+(7)\end{bmatrix}\) = \(\begin{bmatrix}10\\5\\6\\2\end{bmatrix}\)
The multiplication of a column matrix is possible with a row matrix. Satisfying the condition of matrix multiplication, the number of columns in the first matrix should be equal to the number of rows in the second matrix. That is the number of columns in the column matrix for multiplication is equal to the number of rows in the row column.
A = \(\begin{bmatrix}4\\2\\3\\1\end{bmatrix}\), B = \(\begin{bmatrix}7&4&6&5\end{bmatrix}\)
A × B = \(\begin{bmatrix}4\\2\\3\\1\end{bmatrix}\), × \(\begin{bmatrix}7&4&6&5\end{bmatrix}\)
= \(\begin{bmatrix}4×7&4×4&4×6&4×5\\2×7&2×4&2×6&2×5\\3×7&3×4&3×6&3×5\\1×7&1×4&1×6&1×5\end{bmatrix}\)
= \(\begin{bmatrix}28&16&24&20\\14&8&12&10\\21&12&18&15\\7&4&6&5\end{bmatrix}\)
The multiplication of a column matrix with a row matrix results in a square matrix. Further, the column matrix cannot be used for division, since the inverse of a column matrix does not exist.
Related Topics
The following topics help in a better understanding of the column matrix.
Examples on Column Matrix

Example 1: Find the transpose of a column matrix \(\begin{bmatrix}5\\11\\4\\3\end{bmatrix}\).
Solution:
The given matrix is A = \(\begin{bmatrix}5\\11\\4\\3\end{bmatrix}\)
To find the transpose of this column matrix, the column elements are written as row elements.
A^{T} = \(\begin{bmatrix}5&11&4&3\end{bmatrix}\)
Therefore, the transpose of a column matrix is a row matrix.

Example 2: Find the product of the column matrix \(\begin{bmatrix}4 \\5\\3\end{bmatrix}\), and the row matrix \(\begin{bmatrix}2&6&9\end{bmatrix}\).
Solution:
The given matrrices are A = \(\begin{bmatrix}4 \\5\\3\end{bmatrix}\), and B = \(\begin{bmatrix}2&6&9\end{bmatrix}\).
A × B = \(\begin{bmatrix}4 \\5\\3\end{bmatrix}\) × \(\begin{bmatrix}2&6&9\end{bmatrix}\)
= \(\begin{bmatrix}4×2&4×6&4×9\\5×2&5×6&5×9\\3×2&3×6&3×9\\\end{bmatrix}\)
= \(\begin{bmatrix}8&24&36\\10&30&45\\6&18&27\\\end{bmatrix}\)
Therefore, the product of a column matrix and a row matrix is a singleton matrix.
FAQs on Column Matrix
What Is A Column Matrix?
A column matrix is a matrix with only one column, and all the elements are arranged one below the other in a vertical line. The column matrix A = \(\begin{bmatrix}a\\b\\c\\d\end{bmatrix}\), have the four elements placed in a single column. The column matrix has only one column and numerous rows. The order of a column matrix is n × 1.
What Is The Order Of Column Matrix?
The order of a column matrix is n × 1. The column matrix has one column and n number of rows. The number of rows in a column matrix is equal to the number of elements.
What Kind Of A Matrix Is A Column Matrix?
The column matrix is a rectangular matrix. It has an unequal number of rows and columns. The column matrix has one column and numerous rows based on the number of elements in the matrix.
What Is The Transpose Of A Column Matrix?
The transpose of a column matrix gives a row matrix. The column matrix of order n × 1, has a transpose matrix, which is a row matrix of order 1 × n. The column matrix has elements arranged in a vertical manner and the row matrix has elements placed in a vertical format.
What Are The Operations Of A Column Matrix?
The matrix operations of addition, subtraction, and multiplication can be performed using a column matrix. The inverse of a column matrix is not possible since it is not a square matrix. The addition or subtraction of matrices is possible between two column matrices of the same order. The multiplication of a column matrix is possible with a row matrix. Satisfying the condition of matrix multiplication, the number of columns in the column matrix should be equal to the number of rows of a row matrix.
What Is the Difference Between Column Matrix And Row Matrix?
The column matrix has elements arranged in a vertical manner, and the row matrix has elements arranged in a horizontal manner. The order of a column matrix is n × 1, and the order of a row matrix is 1 × n. The column matrix or a row matrix has an equal number of elements. And the product of a column matrix and a row matrix results in a singleton matrix.
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