Row Matrix
A row matrix is a type of matrix that consists of a single row and multiple columns. It is written as:
\[ A = \begin{bmatrix} a_1 & a_2 & a_3 & \dots & a_n \end{bmatrix} \]
where \( A \) has 1 row and \( n \) columns.
Properties of Row Matrices
Property 1: Order of a Row Matrix
The order of a row matrix is always \( 1 \times n \), where \( n \) is the number of columns.
Example
Consider the row matrix:
\[ A = \begin{bmatrix} 5 & -2 & 8 & 3 \end{bmatrix} \]
Since it has 1 row and 4 columns, its order is \( 1 \times 4 \).
Property 2: Addition of Two Row Matrices
Two row matrices of the same order can be added by adding their corresponding elements.
Example
Let:
\[ A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 5 & 6 \end{bmatrix} \]
Then, their sum is:
\[ A + B = \begin{bmatrix} 1+4 & 2+5 & 3+6 \end{bmatrix} = \begin{bmatrix} 5 & 7 & 9 \end{bmatrix} \]
Property 3: Scalar Multiplication
A row matrix can be multiplied by a scalar by multiplying each element by that scalar.
Example
Given:
\[ A = \begin{bmatrix} 2 & -3 & 5 \end{bmatrix} \]
Multiplying by scalar \( k = 3 \):
\[ 3A = \begin{bmatrix} 3 \times 2 & 3 \times (-3) & 3 \times 5 \end{bmatrix} = \begin{bmatrix} 6 & -9 & 15 \end{bmatrix} \]
Property 4: Transpose of a Row Matrix
The transpose of a row matrix is a column matrix. If \( A \) is a row matrix, then its transpose \( A^T \) is:
\[ A^T = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n \end{bmatrix} \]
Example
Given:
\[ A = \begin{bmatrix} 7 & 4 & -1 \end{bmatrix} \]
The transpose is:
\[ A^T = \begin{bmatrix} 7 \\ 4 \\ -1 \end{bmatrix} \]
Property 5: Multiplication with Another Matrix
A row matrix of order \( 1 \times n \) can be multiplied with a matrix of order \( n \times m \) to produce a new matrix of order \( 1 \times m \).
Example
Let:
\[ A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 5 \\ 6 & 7 \\ 8 & 9 \end{bmatrix} \]
The product \( AB \) is:
\[ AB = \begin{bmatrix} (1 \times 4 + 2 \times 6 + 3 \times 8) & (1 \times 5 + 2 \times 7 + 3 \times 9) \end{bmatrix} \]
\[ = \begin{bmatrix} 4 + 12 + 24 & 5 + 14 + 27 \end{bmatrix} = \begin{bmatrix} 40 & 46 \end{bmatrix} \]