Column Matrix

A column matrix is defined as a matrix that has only one column and any number of rows.

• A matrix "A = [a_ij]" is said to be a column matrix if the order of the matrix is "n × 1."
• In a column matrix, all the entries are arranged in a single column.
• A column matrix can have numerous rows but only one column.

Below is the visual representation of the column matrix:

Examples of a Column Matrix

• A column matrix of order "2 × 1" is given below:

A = \left[\begin{array}{c} 6\\ -5 \end{array}\right]_{2\times1}

• A column matrix of order "3 × 1" is given below:

B = \left[\begin{array}{c} 2\\ 1\\ 3 \end{array}\right]_{3\times1}

• A column matrix of order "4 × 1" is given below:

C = \left[\begin{array}{c} 6\\ 4\\ -2\\ 0 \end{array}\right]_{4\times1}

Properties of a Column Matrix

Some important properties of a column matrix are given below,

• Any column matrix will have only one column.
• A column matrix can have numerous rows.
• The number of entries in a column matrix is equal to the number of rows.
• A column matrix is also a rectangular matrix and a vertical matrix.
• The transpose of a column matrix is a row matrix and vice versa.
• Any two-column matrices can be added or subtracted if the order of both matrices is the same.
• The multiplication of a column matrix is possible only with a row matrix if and only if the number of rows in the column matrix is equal to the number of columns in the given row matrix.
• A square matrix is obtained when a column matrix and a row matrix are multiplied.

Operations on Column Matrix

Different algebraic operations, such as addition, subtraction, and multiplication, can be performed on column matrices, but division cannot be performed because its inverse does not exist.

Addition of Column Matrices

Any two-column matrices can be added if the order of both matrices is the same. If the orders of both matrices are the same, then the corresponding entries are added.

For example, \left[\begin{array}{c} 1\\ 0\\ 3 \end{array}\right]_{3\times1}+ \left[\begin{array}{c} 2\\ -4\\ 6 \end{array}\right]_{3\times1}= \left[\begin{array}{c} 3\\ -4\\ 9 \end{array}\right]_{3\times1}

\left[\begin{array}{c} 19\\ 15\\ 21 \end{array}\right]_{3\times1}+ \left[\begin{array}{c} 31\\ 45 \end{array}\right]_{3\times1}⇒ does not exist

Subtraction of Column Matrices

Any two-column matrices can be subtracted if the order of both matrices is the same. If the orders of both matrices are the same, then the corresponding entries are subtracted. 

\left[\begin{array}{c} 12\\ 18\\ -27 \end{array}\right]_{3\times1}- \left[\begin{array}{c} 8\\ -5\\ 0 \end{array}\right]_{3\times1}= \left[\begin{array}{c} 20\\ 23\\ -27 \end{array}\right]_{3\times1}

Multiplication

The multiplication of a column matrix is possible only with a row matrix if and only if the number of rows in the column matrix is equal to the number of columns in the given row matrix. A square matrix is obtained when a column matrix and a row matrix are multiplied.

A × B = \left[\begin{array}{c} 1\\ 3\\ 9 \end{array}\right]_{3\times1}\times\left[\begin{array}{ccc} 2 & 5 & 8\end{array}\right]_{1\times3}

A × B = \left[\begin{array}{ccc} 1\times2 & 1\times5 & 1\times8\\ 3\times2 & 3\times5 & 3\times8\\ 9\times2 & 9\times5 & 9\times8 \end{array}\right]_{3\times3}

A × B = \left[\begin{array}{ccc} 2 & 5 & 8\\ 6 & 15 & 24\\ 18 & 45 & 72 \end{array}\right]_{3\times3}

We can see that the resultant matrix is a square matrix of order "3 × 3."

Column vs Row Matrix

The common difference between row and column matrices is listed in the following table:

Related Articles

• Matrices
• Types of Matrices
• Order of Matrix
• Matrix Operations

Solved Examples on Column Matrix

Example 1: Find the value of Q − 2P, if P = \left[\begin{array}{c} 5\\ 7\\ -9 \end{array}\right]_{3\times1}and Q = \left[\begin{array}{c} 11\\ 21\\ 31 \end{array}\right]_{3\times1} 

Solution:

Q − 2P = \left[\begin{array}{c} 11\\ 21\\ 31 \end{array}\right]_{3\times1} − 2 \times\left[\begin{array}{c} 5\\ 7\\ -9 \end{array}\right]_{3\times1}

Q − 2P = \left[\begin{array}{c} 11\\ 21\\ 31 \end{array}\right]_{3\times1}− \left[\begin{array}{c} 10\\ 14\\ -18 \end{array}\right]_{3\times1}

Q − 2P = \left[\begin{array}{c} (11-10)\\ (21-14)\\ (31-(-18)) \end{array}\right]_{3\times1}

Q − 2P = \left[\begin{array}{c} 1\\ 7\\ 49 \end{array}\right]_{3\times1}

Example 2: Prove that the transpose of a column matrix is a row matrix.

Solution:

Let us consider an example, to prove that the transpose of a column matrix is a row matrix.

A = \left[\begin{array}{c} 15\\ 0\\ -13 \end{array}\right]_{3\times1}

The above matrix is a column matrix of order "3 × 1." We know that the transpose of a matrix is obtained by interchanging the entries of rows and columns. So, the order of the transpose of the given matrix will be "1 × 3."

A = \left[\begin{array}{c} 15\\ 0\\ -13 \end{array}\right]_{3\times1}⇒ A^{T} = \left[\begin{array}{ccc} 15 & 0 & -13\end{array}\right]_{1\times3}

We can see that the resultant matrix is a row matrix.

Hence proved.

Example 3: Find the product of the matrices given below.

A = \left[\begin{array}{c} 4\\ -5\\ 6 \end{array}\right]_{3\times1} and B = \left[\begin{array}{ccc} 1 & 0 & 2\end{array}\right]_{1\times3}

Solution:

A × B = \left[\begin{array}{c} 4\\ -5\\ 6 \end{array}\right]_{3\times1}\times\left[\begin{array}{ccc} 1 & 0 & 2\end{array}\right]_{1\times3}

A × B = \left[\begin{array}{ccc} 4\times1 & 4\times0 & 4\times2\\ -5\times1 & -5\times0 & -5\times2\\ 6\times1 & 6\times0 & 6\times2 \end{array}\right]_{3\times3}

A × B = \left[\begin{array}{ccc} 4 & 0 & 8\\ -5 & 0 & -10\\ 6 & 0 & 12 \end{array}\right]_{3\times3}

Example 4: Find the value of M − N, if 

M = \left[\begin{array}{c} 81\\ 72\\ 63 \end{array}\right]_{3\times1} and N = \left[\begin{array}{c} -48\\ 36\\ 21 \end{array}\right]_{3\times1}

Solution:

M - N = \left[\begin{array}{c} 81\\ 72\\ 63 \end{array}\right]_{3\times1}- \left[\begin{array}{c} -48\\ 36\\ 21 \end{array}\right]_{3\times1}

M - N = \left[\begin{array}{c} 81+48\\ 72-36\\ 63-21 \end{array}\right]_{3\times1}

M - N = \left[\begin{array}{c} 129\\ 36\\ 42 \end{array}\right]_{3\times1}