Question

If a real matrix \(A\) satisfies \[ A^T=A \quad \text{and} \quad A^2=I, \] then the eigenvalues of \(A\) must be:

• A. Only \(1\)
• B. Only \(-1\)
• C. Either \(1\) or \(-1\)
• D. Zero only

Hint:

Whenever a matrix equation like \[ A^2=I \] appears, immediately convert it into an eigenvalue equation: \[ \lambda^2=1 \] This simplifies the problem instantly.

Answer 1

The Correct Option is C

Concept: If \(\lambda\) is an eigenvalue of matrix \(A\), then: \[ A\mathbf{x}=\lambda \mathbf{x} \] Applying powers of matrices transforms eigenvalues similarly.

Step 1: Use the matrix condition \(A^2=I\).
Suppose \(\lambda\) is an eigenvalue of \(A\). Then: \[ A\mathbf{x}=\lambda \mathbf{x} \] Applying \(A\) again: \[ A^2\mathbf{x}=\lambda^2\mathbf{x} \] But: \[ A^2=I \] Hence: \[ I\mathbf{x}=\lambda^2\mathbf{x} \] \[ \lambda^2=1 \]

Step 2: Solve for possible eigenvalues.
\[ \lambda^2=1 \] Thus: \[ \lambda=\pm1 \] Therefore, the eigenvalues must be: \[ \boxed{1 \text{ or } -1} \]