Question

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

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

Hint:

Always check the diagonal and off-diagonal elements when identifying an identity matrix.

Answer 1

The Correct Option is D

Step 1: Identity Matrix Definition
An identity matrix, denoted as \( A = [a_{ij}] \), is a square matrix characterized by having 1s along its main diagonal and 0s everywhere else.
Step 2: Conditions for \( a_{ij} \)
The elements \( a_{ij} \) of an identity matrix satisfy the following conditions: \[ a_{ij} = \begin{cases} 0, & \text{if } i eq j \\ 1, & \text{if } i = j \end{cases} \] Step 3: Option Verification
Option (D) aligns precisely with this definition.