Topic: Engineering Mathematics
For the matrix $\begin{bmatrix} 4 & 1 \\ 1 & 4 \end{bmatrix}$, the eigen values are
- A. 3 and -3
- B. -3 and -5
- C. 3 and 5
- D. 5 and 0
Answer: C. 3 and 5
Explanation: Let the eigenvalues be $\lambda_1$ and $\lambda_2$. The sum of the eigenvalues equals the trace of the matrix, so $\lambda_1 + \lambda_2 = 8$. The product of the eigenvalues equals the determinant, so $\lambda_1 \cdot \lambda_2 = 15$. The quadratic equation for eigenvalues is $\lambda² - 8\lambda + 15 = 0$. Factoring this equation provides $(\lambda - 5)(\lambda - 3) = 0$, yielding the eigenvalues 3 and 5.