Question

Pick the CORRECT eigenvalue(s) of the matrix [A] from the following choices.

\[ [A] = \begin{bmatrix} 6 & 8 \\ 4 & 2 \end{bmatrix} \]

• A. 10
• B. 4
• C. -2
• D. -10

Hint:

To find the eigenvalues of a matrix, solve the characteristic equation \( {det}(A - \lambda I) = 0 \). This gives a polynomial, and the roots of that polynomial are the eigenvalues.

Answer 1

The Correct Option is A, C

Given matrix: \[ A = \begin{bmatrix} 6 & 8 \\ 4 & 2 \end{bmatrix} \]
Step 1: Find the characteristic equation
The characteristic equation is given by: \[ \det(A - \lambda I) = 0 \] Substituting \( A \): \[ \begin{vmatrix} 6 - \lambda & 8 \\ 4 & 2 - \lambda \end{vmatrix} = 0 \]
Step 2: Compute determinant
Expanding the determinant: \[ (6 - \lambda)(2 - \lambda) - (8)(4) = 0 \] \[ (6 - \lambda)(2 - \lambda) = 32 \] Expanding: \[ 12 - 6\lambda - 2\lambda + \lambda^2 = 32 \] \[ \lambda^2 - 8\lambda - 20 = 0 \]
Step 3: Solve for \( \lambda \)
Factorizing: \[ \lambda^2 - 10\lambda + 2\lambda - 20 = 0 \] \[ \lambda(\lambda - 10) + 2(\lambda - 10) = 0 \] \[ (\lambda - 10)(\lambda + 2) = 0 \]
Step 4: Eigenvalues
Solving for \( \lambda \): \[ \lambda = -2, 10 \]
Final Answer: \[ \boxed{-2, 10} \]