Question

Let \(A\) be an \(n \times n\) matrix of real number such that \(A^2 = A\), then the Eigen values \(A\) are:

• A. 0, 1
• B. 1, -1
• C. 1, -1
• D. -1, -1

Hint:

To find the possible eigenvalues of any matrix defined by a polynomial equation \(p(A) = 0\), simply solve the polynomial equation \(p(\lambda) = 0\) using ordinary algebra.
For \(A^2 - A = 0 \implies \lambda^2 - \lambda = 0 \implies \lambda = 0, 1\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A matrix \(A\) that satisfies the equation \(A^2 = A\) is called an idempotent matrix.
Eigenvalues \(\lambda\) of any square matrix must satisfy its matrix polynomial equation.

Step 2: Key Formula or Approach:
The eigenvalue equation is:
\[ A v = \lambda v \]
where \(\lambda\) is the eigenvalue and \(v\) is a non-zero eigenvector.

Step 3: Detailed Explanation:
Let \(\lambda\) be an eigenvalue of the idempotent matrix \(A\), and let \(v \neq 0\) be its corresponding eigenvector.
By definition:
\[ A v = \lambda v \]
Now, let us multiply both sides of this equation by the matrix \(A\) from the left:
\[ A(Av) = A(\lambda v) \]
Using the properties of matrix multiplication, we rewrite this as:
\[ A^2 v = \lambda (Av) \]
Substitute \(Av = \lambda v\) into the right-hand side of the equation:
\[ A^2 v = \lambda (\lambda v) = \lambda^2 v \]
Since \(A\) is idempotent, we substitute \(A^2 = A\) into the left-hand side:
\[ A v = \lambda^2 v \]
Again, using the definition \(Av = \lambda v\), we get:
\[ \lambda v = \lambda^2 v \]
Rearranging this vector equation:
\[ \lambda^2 v - \lambda v = 0 \]
\[ (\lambda^2 - \lambda) v = 0 \]
Since \(v\) is an eigenvector, it must be a non-zero vector (\(v \neq 0\)).
Therefore, the scalar term must equal zero:
\[ \lambda^2 - \lambda = 0 \]
Factoring the quadratic equation:
\[ \lambda(\lambda - 1) = 0 \]
This gives two possible solutions for the eigenvalue:
\[ \lambda = 0 \quad \text{or} \quad \lambda = 1 \]
Thus, the eigenvalues of an idempotent matrix can only be 0 or 1.

Step 4: Final Answer:
Therefore, the correct option is (A).