Step 1: Definition of an identity matrix.
An identity matrix \( A = [a_{ij}] \) is a square matrix in which all the diagonal elements are \( 1 \), and all off-diagonal elements are \( 0 \). Mathematically:
\[
a_{ij} =
\begin{cases}
1, & {if } i = j,
0, & {if } i \neq j.
\end{cases}
\]
Step 2: Analyze each option.
- (A) \( a_{ij} = 0 { if } i = j { and } a_{ij} = 1 { if } i \neq j \): This is incorrect because it contradicts the definition of an identity matrix.
- (B) \( a_{ij} = 1, \, \forall i, j \): This is incorrect because an identity matrix has \( 0 \) for all off-diagonal elements.
- (C) \( a_{ij} = 0, \, \forall i, j \): This is incorrect because it implies all elements are \( 0 \), which is not an identity matrix.
- (D) \( a_{ij} = 0 { if } i \neq j { and } a_{ij} = 1 { if } i = j \): This is correct, as it matches the definition of an identity matrix.
Final Answer: \( \boxed{ {(D)}} \)