Question

A matrix which, when multiplied by itself, yields itself is known as

• A. Symmetric matrix
• B. Nilpotent matrix
• C. Idempotent matrix
• D. Hermitian matrix

Hint:

The word "Idempotent" comes from "idem" (meaning same) and "potence" (meaning power). Raising the matrix to any power yields the same matrix:
\[ A^n = A \quad \forall n \ge 1 \]

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Special types of matrices are defined based on properties they exhibit when subjected to operations like transposition, conjugate transposition, or multiplication.

Step 2: Detailed Explanation:
Let us review the definitions of each option to clarify why Option (C) is correct:
- Option (A) Symmetric matrix: A square matrix \(A\) is symmetric if it is equal to its transpose:
\[ A^T = A \] This property is defined by transposition, not multiplication.
- Option (B) Nilpotent matrix: A square matrix \(A\) is nilpotent if raising it to some positive integer power \(k\) results in the zero matrix:
\[ A^k = 0 \] This is the opposite of yielding itself.
- Option (C) Idempotent matrix: A square matrix \(A\) is idempotent if, when multiplied by itself, it yields the original matrix:
\[ A^2 = A \] This matches the criteria in the question.
- Option (D) Hermitian matrix: A complex square matrix \(A\) is Hermitian if it is equal to its conjugate transpose:
\[ A^* = A \] This relates to complex conjugation and transposition.

Step 3: Final Answer:
The correct option is (C).