A matrix that consists of all elements in only one column is called a column matrix.

A column matrix is one type of matrix. In this matrix, the elements are arranged in a number of rows and but in one column. Hence, it is called a column matrix and also called as a column vector.

For example, we have some elements but all elements are arranged in only one column. The elements are actually arranged in different rows for separating them. Let’s assume that all elements are arranged in $m$ rows in this case. Therefore, the elements are arranged in $m$ row and $1$ column.

A matrix $M$ of the order $m \times 1$ is formed and it can be written mathematically in the following form.

$M$ $\,=\,$ ${\begin{bmatrix} e_{11}\\ e_{21}\\ e_{31}\\ \vdots\\ e_{m1} \end{bmatrix}}_{\displaystyle m \times 1} $

The arrangement of elements in this matrix represents a rectangle shape. So, a column matrix is actually a rectangular matrix and it is simply expressed as follows.

$M = {\begin{bmatrix} e_{\displaystyle ij} \end{bmatrix}}_{\displaystyle m \times n} $

In the case of a column vector, the number of columns $j = 1$. So, $n = 1$ but $i = m$. Hence, the simple form of a column matrix can be written in the following matrix form.

$M = {\begin{bmatrix} e_{\displaystyle i1} \end{bmatrix}}_{\displaystyle m \times 1} $

The following matrices are best examples for a column matrix.

$(1).\,\,\,\,$ $A = \begin{bmatrix} 7 \end{bmatrix} $

$A$ is a column matrix of the order $1 \times 1$. In this column matrix, the only one element is displayed in one row and one column.

$(2).\,\,\,\,$ $B = \begin{bmatrix} -1\\ 4 \end{bmatrix} $

$B$ is a column matrix of the order $2 \times 1$ and in this matrix, the two elements are arranged in two rows and one column.

$(3).\,\,\,\,$ $C = \begin{bmatrix} 6\\ 0\\ 9 \end{bmatrix} $

$C$ is a column matrix of the order $3 \times 1$. The three elements are arranged in the matrix in three rows and one column.

$(4).\,\,\,\,$ $D = \begin{bmatrix} -5\\ 8\\ 2\\ 3 \end{bmatrix} $

$D$ is a column matrix of the order $4 \times 1$. The four elements are arranged in the matrix in four rows and one column.

In all above four examples, the elements are arranged in only one column but the number of rows are different. Actually, the arrangement of elements in all matrices forms a rectangle shape. So, a column matrix is always a rectangular matrix.

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