Column Matrix
A column matrix is a type of matrix that has only one column and multiple rows. It is represented as:
\[ A = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n \end{bmatrix} \]
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:
\[ A = \begin{bmatrix} 5 \\ -2 \\ 7 \\ 3 \end{bmatrix} \]
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:
\[ A = \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix}, \quad B = \begin{bmatrix} 1 \\ -3 \\ 5 \end{bmatrix} \]
Then, the sum is:
\[ A + B = \begin{bmatrix} 2+1 \\ 4+(-3) \\ 6+5 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 11 \end{bmatrix} \]
3 Scalar Multiplication
A column matrix can be multiplied by a scalar (a single number) by multiplying each element by the scalar.
Example
Let:
\[ A = \begin{bmatrix} 3 \\ -1 \\ 4 \end{bmatrix}, \quad k = 2 \]
Then, the scalar multiplication is:
\[ kA = 2 \times \begin{bmatrix} 3 \\ -1 \\ 4 \end{bmatrix} = \begin{bmatrix} 6 \\ -2 \\ 8 \end{bmatrix} \]
4 Transpose of a Column Matrix
The transpose of a column matrix converts it into a row matrix.
Example
Let:
\[ A = \begin{bmatrix} 7 \\ 8 \\ -5 \end{bmatrix} \]
Then, its transpose is:
\[ A^T = \begin{bmatrix} 7 & 8 & -5 \end{bmatrix} \]
5 Multiplication of Two Column Matrices
Two column matrices cannot be directly multiplied unless one is transposed to form a row matrix.
Example
Let:
\[ A = \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \quad B = \begin{bmatrix} -1 \\ 4 \end{bmatrix} \]
To perform matrix multiplication, we take the transpose of \( A \):
\[ A^T B = \begin{bmatrix} 2 & 3 \end{bmatrix} \times \begin{bmatrix} -1 \\ 4 \end{bmatrix} \]
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:
\[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 5 \\ -2 \\ 3 \end{bmatrix} \]
Then:
\[ IA = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} 5 \\ -2 \\ 3 \end{bmatrix} = \begin{bmatrix} 5 \\ -2 \\ 3 \end{bmatrix} \]
7 Zero Column Matrix
A column matrix in which all elements are zero is called a zero column matrix.
Example
\[ Z = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]
For any column matrix \( A \), adding \( Z \) does not change \( A \):
\[ A + Z = A \]