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

Quadratic formula with a=1,b=0,c=-25 gives lambda = plus-or-minus 5. Verify with eigenvector for lambda=5: solve (A-5I)v=0, giving v=(1,2). Check Av = (5,10) = 5v, confirming lambda=5 is genuine; the other eigenvalue is -5 since eigenvalues sum to the trace, 0.

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