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

Options

A. Horizontal matrix if `mgtn`
B. Horizontal matrix if `mltn`
C. Vertical matrix if `mgtn`
D. Vertical matrix if `mltn`

Answer

To solve the problem regarding the classification of a matrix \( A = [a_{ij}]_{m \times n} \) based on the number of rows \( m \) and columns \( n \), we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Matrix Dimensions**:
- A matrix \( A \) is defined as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.

2. **Identifying a Horizontal Matrix**:
- A matrix is called a **horizontal matrix** if the number of rows is less than the number of columns, i.e., \( m < n \).
- Therefore, if \( m > n \), the matrix cannot be horizontal.

3. **Analyzing Option 1**:
- **Option 1** states: "Horizontal matrix if \( m > n \)".
- This statement is **false** because for a horizontal matrix, we require \( m < n \).

4. **Identifying a Vertical Matrix**:
- A matrix is called a **vertical matrix** if the number of rows is greater than the number of columns, i.e., \( m > n \).

5. **Analyzing Option 2**:
- **Option 2** states: "Horizontal matrix if \( m < n \)".
- This statement is **true** because it correctly identifies the condition for a horizontal matrix.

6. **Analyzing Option 3**:
- **Option 3** states: "Vertical matrix if \( m > n \)".
- This statement is **true** because it correctly identifies the condition for a vertical matrix.

7. **Analyzing Option 4**:
- **Option 4** states: "Vertical matrix if \( m < n \)".
- This statement is **false** because for a vertical matrix, we require \( m > n \).

### Conclusion:
- The correct options based on the definitions are:
- **Option 2**: True (Horizontal matrix if \( m < n \))
- **Option 3**: True (Vertical matrix if \( m > n \))

### Final Answer:
- The correct options are **Option 2** and **Option 3**.