Question

The eigenvalue(s) of the matrix 

• A. \(-1\)
• B. 1
• C. 2
• D. 3

Hint:

The eigenvalues of a 2x2 matrix can be found by solving the characteristic equation: \(\text{det}(A - \lambda I) = 0\).

Answer 1

The Correct Option is A, D

Step 1: Find the characteristic equation.
For the matrix \(\begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}\), we calculate the determinant of \(\begin{bmatrix} 1-\lambda & 2 \\ 2 & 1-\lambda \end{bmatrix}\). The characteristic equation is: \[ \text{det} \begin{bmatrix} 1-\lambda & 2 \\ 2 & 1-\lambda \end{bmatrix} = (1-\lambda)^2 - 4 = 0 \] \[ (1-\lambda)^2 = 4 \quad \Rightarrow \quad \lambda = 1 \text{ or } \lambda = -1 \] Step 2: Conclusion.
The eigenvalues are 1 and -1, and hence, the correct answer is (B) 1.