Column Matrix and Its Properties

Column Matrix

A column matrix is a type of matrix that has only one column and multiple rows. It is represented as:

$$A=\left[\begin{array}{c}{a}_1\\ a_2\\ a_3\\ ⋮\\ a_n\end{array}\right]$$

where $A$ is an $n\times 1$ matrix, meaning it has $n$ rows and 1 column.

Properties of a Column Matrix

1 Order of a Column Matrix

A column matrix is always of the order $n\times 1$, where $n$ represents the number of rows.

Example

Consider a 4×1 column matrix:

$$A=\left[\begin{array}{c}5\\ -2\\ 7\\ 3\end{array}\right]$$

Here, the matrix has 4 rows and 1 column, so its order is $4\times 1$.

2 Addition of Column Matrices

Two column matrices of the same order can be added by adding their corresponding elements.

Example

Let:

$$A=\left[\begin{array}{c}2\\ 4\\ 6\end{array}\right],B=\left[\begin{array}{c}1\\ -3\\ 5\end{array}\right]$$

Then, the sum is:

$$A+B=\left[\begin{array}{c}2+1\\ 4+(-3)\\ 6+5\end{array}\right]=\left[\begin{array}{c}3\\ 1\\ 11\end{array}\right]$$

3 Scalar Multiplication

A column matrix can be multiplied by a scalar (a single number) by multiplying each element by the scalar.

Example

Let:

$$A=\left[\begin{array}{c}3\\ -1\\ 4\end{array}\right],k=2$$

Then, the scalar multiplication is:

$$kA=2\times \left[\begin{array}{c}3\\ -1\\ 4\end{array}\right]=\left[\begin{array}{c}6\\ -2\\ 8\end{array}\right]$$

4 Transpose of a Column Matrix

The transpose of a column matrix converts it into a row matrix.

Example

Let:

$$A=\left[\begin{array}{c}7\\ 8\\ -5\end{array}\right]$$

Then, its transpose is:

$$A^T=\left[\begin{array}{ccc}7& 8& -5\end{array}\right]$$

5 Multiplication of Two Column Matrices

Two column matrices cannot be directly multiplied unless one is transposed to form a row matrix.

Example

Let:

$$A=\left[\begin{array}{c}2\\ 3\end{array}\right],B=\left[\begin{array}{c}-1\\ 4\end{array}\right]$$

To perform matrix multiplication, we take the transpose of $A$:

$$A^{T}B=\left[\begin{array}{cc}2& 3\end{array}\right]\times \left[\begin{array}{c}-1\\ 4\end{array}\right]$$

The result is:

$$A^{T}B=(2\times -1)+(3\times 4)=-2+12=10$$

6 Identity Property

Multiplying a column matrix by an identity matrix of compatible order results in the original matrix.

Example

Let:

$$I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],A=\left[\begin{array}{c}5\\ -2\\ 3\end{array}\right]$$

Then:

$$IA=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\times \left[\begin{array}{c}5\\ -2\\ 3\end{array}\right]=\left[\begin{array}{c}5\\ -2\\ 3\end{array}\right]$$

7 Zero Column Matrix

A column matrix in which all elements are zero is called a zero column matrix.

Example

$$Z=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$$

For any column matrix $A$, adding $Z$ does not change $A$:

$$A+Z=A$$