Question

A matrix \( B = [b_{ij}]_{m \times m} \) is said to be a diagonal matrix, if :

• A. \( b_{ij} = 0 \), when \( i = j \)
• B. \( b_{ij} = 1 \), when \( i = j \)
• C. \( b_{ij} = 1 \), when \( i \neq j \)
• D. \( b_{ij} = 0 \), when \( i \neq j \)

Hint:

A scalar matrix is a special diagonal matrix where all diagonal elements are equal (\( b_{ii} = k \)).
An identity matrix is a special diagonal matrix where all diagonal elements are equal to 1 (\( b_{ii} = 1 \)).

Answer 1

The Correct Option is D

Concept:
• A diagonal matrix is a square matrix where all the elements outside the main diagonal are zero.
• The main diagonal elements are those where the row index equals the column index (\( i = j \)).
• Non-diagonal elements are those where the row index does not equal the column index (\( i \neq j \)).

Step 1: Identify the condition for diagonal elements
In a matrix \( B = [b_{ij}]_{m \times m} \), the elements \( b_{11}, b_{22}, \dots, b_{mm} \) are diagonal elements.
For these elements, the condition is \( i = j \).
In a diagonal matrix, these elements can be any value (zero or non-zero).

Step 2: Identify the condition for non-diagonal elements
All other elements in the matrix are called non-diagonal elements.
For these elements, the condition is \( i \neq j \).
By definition, for a matrix to be diagonal, every non-diagonal element must be zero.
\[ b_{ij} = 0 \text{ for all } i \neq j \]