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

Options

A. `I lt i`
B. `i=j`
C. `I gt j`
D. `I lt j`

Answer

To determine the condition for a matrix \( A = [a_{ij}] \) to be classified as an upper triangular matrix, we can follow these steps:

### Step 1: Understand the Definition of Upper Triangular Matrix
An upper triangular matrix is defined as a square matrix in which all the entries below the main diagonal are zero. This means that for any element \( a_{ij} \) in the matrix, if the row index \( i \) is greater than the column index \( j \) (i.e., \( i > j \)), then \( a_{ij} \) must be equal to zero.

### Step 2: Analyze the Matrix Structure
Consider a square matrix of order \( n \times n \). The main diagonal consists of elements where the row index equals the column index (i.e., \( a_{ii} \)). The elements below this diagonal (where \( i > j \)) must all be zero for the matrix to be upper triangular.

### Step 3: Formulate the Condition
From the definition, we can express the condition mathematically:
- For an upper triangular matrix \( A = [a_{ij}] \):
\[ a_{ij} = 0 \quad \text{for all } i > j \]

### Step 4: Conclusion
Thus, the correct statement that defines an upper triangular matrix is:
- A matrix \( A = [a_{ij}] \) is an upper triangular matrix if \( a_{ij} = 0 \) whenever \( i > j \).

### Final Answer
The correct option is that a matrix \( A = [a_{ij}] \) is an upper triangular matrix if \( a_{ij} = 0 \) for all \( i > j \).

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