Question

Which one of the following is the pair of eigenvalues of the matrix \( \begin{bmatrix} -3 & 4 \\ 4 & 3 \end{bmatrix} \)?

• A. \( -7, 7 \)
• B. \( -5, 5 \)
• C. \( -4, 3 \)
• D. \( -3, 4 \)

Hint:

Use lambda^2 - (trace)lambda + det(A) = 0; here trace = 0 and det = -25, giving lambda^2 = 25.

Answer 1

The Correct Option is B

Step 1: Characteristic equation. \( \det(A-\lambda I)=(-3-\lambda)(3-\lambda)-16=\lambda^2-25=0 \), giving \( \lambda=\pm5 \).

Step 2: Verify using trace and determinant. trace(A)=0, det(A)=-25. For lambda=-5,5: sum=0 (matches trace), product=-25 (matches determinant).

Step 3: Eliminate other options via the same checks. -7,7 fails determinant (-49 not -25). -4,3 and -3,4 both fail the trace check (sum not zero).

\[ \boxed{\lambda = -5,\ 5 \ \Rightarrow \ \text{Option (B)}} \]