Question

If \( A = [a_{ij}] \) is an identity matrix, then which of the following is true?

• A. \( a_{ij} = 0, & {if } i = j,
  1, & {if } i \neq j \)
• B. \( a_{ij} = 1, \, \forall i, j \)
• C. \( a_{ij} = 0, \, \forall i, j \)
• D. \( a_{ij} = 0, & {if } i \neq j,
  1, & {if } i = j \)

Hint:

For identity matrices: \item Diagonal elements are always \( 1 \). \item Off-diagonal elements are always \( 0 \).

Answer 1

The Correct Option is D

Step 1: Define an identity matrix.
An identity matrix \( A = [a_{ij}] \) is a square matrix where diagonal elements are \( 1 \) and off-diagonal elements are \( 0 \). This is expressed mathematically as:\[a_{ij} = \begin{cases} 1, & {if } i = j,
0, & {if } i eq j.\end{cases}\]Step 2: Evaluate each option.
- (A) \( a_{ij} = 0 { if } i = j { and } a_{ij} = 1 { if } i eq j \): This option is incorrect as it contradicts the definition of an identity matrix.- (B) \( a_{ij} = 1, \, \forall i, j \): This option is incorrect because an identity matrix must have \( 0 \) for all off-diagonal elements.- (C) \( a_{ij} = 0, \, \forall i, j \): This option is incorrect because it implies all elements are \( 0 \), which does not constitute an identity matrix.- (D) \( a_{ij} = 0 { if } i eq j { and } a_{ij} = 1 { if } i = j \): This option is correct, aligning with the established definition of an identity matrix.
Final Answer: \( \boxed{ {(D)}} \)