Question

Let $A$ be a square matrix. If $A^2 = A$, then the matrix $A$ is called:

• A. Nilpotent
• B. Idempotent
• C. Involutory
• D. Singular

Hint:

Remember these key matrix properties: $A^2 = A$ (Idempotent), $A^2 = I$ (Involutory), $A^k = 0$ (Nilpotent).

Answer 1

The Correct Option is B

Step 1: Understanding the given condition.
The condition given in the question is $A^2 = A$. This means that when the matrix $A$ is multiplied by itself, the result is the same matrix $A$.
Step 2: Definition of an idempotent matrix.
A square matrix $A$ is called an idempotent matrix if it satisfies the condition \[ A^2 = A \] This definition directly matches the given condition in the question.
Step 3: Analysis of the given options.
(A) Nilpotent: A nilpotent matrix satisfies $A^k = 0$ for some positive integer $k$, which is not given here.
(B) Idempotent: Correct — an idempotent matrix satisfies $A^2 = A$.
(C) Involutory: An involutory matrix satisfies $A^2 = I$, where $I$ is the identity matrix.
(D) Singular: A singular matrix is one whose determinant is zero, which is unrelated to the given condition.
Step 4: Conclusion.
Since the matrix satisfies $A^2 = A$, it is correctly classified as an idempotent matrix.