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: Definition of an identity matrix
An identity matrix \( A = [a_{ij}] \) is defined as a square matrix where the diagonal elements are 1 and all other elements are 0.
Step 2: Express the conditions for \( a_{ij} \)
For \( a_{ij} \) in an identity matrix: \[ a_{ij} = \begin{cases} 0, & \text{if } i \neq j \\ 1, & \text{if } i = j \end{cases} \] Step 3: Verify the options
Option (D) correctly matches this definition.