A matrix `A=[a_(ij)]` is an upper triangular matrix, if

Options

A. it is a square matrix and `a_(ij) =0,iltj`
B. it is a square matrix and `a_(ij) =0,igtj`
C. it is not a square matrix and `a_(ij)=0,igtj`
D. it is not a square matrix and `a_(ij) =0,iltj`

Answer

To determine the conditions under which a matrix \( A = [a_{ij}] \) is classified as an upper triangular matrix, we can follow these steps:

### Step-by-Step Solution:

1. **Definition of Upper Triangular Matrix**:
An upper triangular matrix is defined as a square matrix where all the entries below the main diagonal are zero. This means that for a matrix \( A \) of size \( n \times n \), the elements \( a_{ij} \) must satisfy certain conditions based on their positions.

2. **Matrix Structure**:
Consider a matrix \( A \) of size \( n \times n \):
\[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn} \end{bmatrix} \]

3. **Condition for Upper Triangular Matrix**:
For \( A \) to be an upper triangular matrix, the following condition must hold:
\[ a_{ij} = 0 \quad \text{for all } i > j \]
This means that if the row index \( i \) is greater than the column index \( j \), the corresponding element \( a_{ij} \) must be zero.

4. **Examples of Upper Triangular Matrices**:
- A \( 2 \times 2 \) upper triangular matrix:
\[ A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \]
- A \( 3 \times 3 \) upper triangular matrix:
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & -1 & 4 \\ 0 & 0 & 5 \end{bmatrix} \]

5. **Conclusion**:
Therefore, a matrix \( A = [a_{ij}] \) is an upper triangular matrix if all the elements below the main diagonal are zero, which can be mathematically expressed as \( a_{ij} = 0 \) for all \( i > j \).