Question

Which one of the following matrices has eigenvalues 1 and 6?

• A. $\begin{bmatrix} 5 & -2 \\ -2 & 2 \end{bmatrix}$
• B. $\begin{bmatrix} 3 & -1 \\ -2 & 2 \end{bmatrix}$
• C. $\begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}$
• D. $\begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}$

Hint:

When solving for eigenvalues, remember to carefully calculate the determinant of $(A - \lambda I)$ and solve for $\lambda$. The eigenvalues are the solutions to the resulting characteristic polynomial.

Answer 1

The Correct Option is A

To determine the correct matrix, we need to find the eigenvalues of each matrix by solving the characteristic equation ${\det}(A - \lambda I) = 0$, where $\lambda$ is the eigenvalue and $I$ is the identity matrix. 
Step 1: Eigenvalue calculation for each matrix The characteristic equation for a matrix $A$ is derived from the determinant of $A - \lambda I$. Once we compute this determinant, we find the eigenvalues by solving for $\lambda$.

Step 2: Analyze the options
Option (A): For the matrix $\begin{bmatrix} 5 & -2 \\ -2 & 2 \end{bmatrix}$, we compute the determinant of $(A - \lambda I)$ and find that the eigenvalues are indeed 1 and 6, which is exactly what the question is asking for. Hence, Option (A) is the correct answer.

Option (B): The matrix $\begin{bmatrix} 3 & -1 \\ -2 & 2 \end{bmatrix}$ has eigenvalues that do not match 1 and 6. Solving for the eigenvalues gives us different results, so this option is incorrect.

Option (C): The matrix $\begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}$ also has eigenvalues that do not match the required 1 and 6. While solving, we find eigenvalues that are different from the target. This option is incorrect.

Option (D): The matrix $\begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}$ gives eigenvalues that do not match 1 and 6 either. The eigenvalue calculation shows results that do not satisfy the condition. Thus, Option (D) is incorrect.

Step 3: Conclusion
Based on the calculation of eigenvalues, we conclude that only Option (A) correctly matches the required eigenvalues of 1 and 6.