A matrix `A=[a_(ij)]_(mxxn)` is

Options

A. (a)Horizontal matrix if `mgtn`
B. (b)Horizontal matrix if `mltn`
C. (c)Vertical matrix if `mgtn`
D. (d)Vertical matrix if `mltn`

Answer

To solve the question regarding the matrix \( A = [a_{ij}]_{m \times n} \), we need to analyze the conditions given for horizontal and vertical matrices.

### Step-by-Step Solution:

1. **Understanding the Matrix Dimensions**:
- A matrix \( A \) of order \( m \times n \) has \( m \) rows and \( n \) columns.
- Here, \( m \) represents the number of rows and \( n \) represents the number of columns.

2. **Identifying Horizontal Matrix**:
- A horizontal matrix is one where the number of columns \( n \) is greater than the number of rows \( m \) (i.e., \( m < n \)).
- For example, consider a matrix:
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \]
This matrix has 2 rows and 3 columns, hence it is of order \( 2 \times 3 \) (where \( m < n \)).

3. **Identifying Vertical Matrix**:
- A vertical matrix is one where the number of rows \( m \) is greater than the number of columns \( n \) (i.e., \( m > n \)).
- For example, consider a matrix:
\[ A = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} \]
This matrix has 3 rows and 2 columns, hence it is of order \( 3 \times 2 \) (where \( m > n \)).

4. **Conclusion**:
- For a horizontal matrix, the condition is \( m < n \), which corresponds to option B.
- For a vertical matrix, the condition is \( m > n \), which corresponds to option C.

### Final Answer:
- The correct option for a horizontal matrix is **Option B** (where \( m < n \)).
- The correct option for a vertical matrix is **Option C** (where \( m > n \)).