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: Interpreting the given relation.  
The condition stated in the problem is $A^2 = A$. This indicates that multiplying the matrix $A$ by itself yields the same matrix $A$. 

Step 2: Relevant matrix definition. 
A square matrix is said to be idempotent if it satisfies the relation \[ A^2 = A \] This definition exactly matches the condition provided in the question. 

Step 3: Evaluation of the options. 
(A) Nilpotent: A nilpotent matrix satisfies $A^k = 0$ for some positive integer $k$, which does not apply here. 
(B) Idempotent: Correct, since an idempotent matrix is defined by $A^2 = A$. 
(C) Involutory: An involutory matrix satisfies $A^2 = I$, where $I$ denotes the identity matrix, which is different from the given condition. 
(D) Singular: A singular matrix has a determinant equal to zero; this property is not implied by the given relation. 

Step 4: Final inference. 
Because the matrix fulfills the condition $A^2 = A$, it is correctly identified as an idempotent matrix.