Question

Which of the following is one of the eigenvalues for the matrix given below?
\[ \begin{bmatrix} 3 & 4 \\ 4 & -3 \end{bmatrix} \]

• A. 1
• B. 3
• C. 5
• D. 7

Hint:

For a 2x2 matrix, eigenvalues satisfy sum = trace and product = determinant; here trace=0 and determinant=-25.

Answer 1

The Correct Option is C

Step 1: Understanding the Question.
We need one eigenvalue of the \(2\times2\) matrix
\[ A = \begin{bmatrix} 3 & 4 \\ 4 & -3 \end{bmatrix} \]

Step 2: Key Formula or Approach.
Eigenvalues \(\lambda\) of a matrix \(A\) satisfy the characteristic equation \(\det(A - \lambda I) = 0\). For a \(2\times2\) matrix \(\begin{bmatrix} p & q \\ r & s \end{bmatrix}\), this becomes \((p-\lambda)(s-\lambda) - qr = 0\).

Step 3: Detailed Explanation.
Here \(p=3\), \(q=4\), \(r=4\), \(s=-3\). Set up the characteristic equation:
\[ (3-\lambda)(-3-\lambda) - (4)(4) = 0 \]
Expand \((3-\lambda)(-3-\lambda)\):
\[ (3-\lambda)(-3-\lambda) = -9 -3\lambda +3\lambda +\lambda^2 = \lambda^2 - 9 \]
So the equation becomes:
\[ \lambda^2 - 9 - 16 = 0 \]
\[ \lambda^2 = 25 \]
\[ \lambda = \pm 5 \]
So the two eigenvalues of \(A\) are \(5\) and \(-5\).
Check option (C), \(5\): it matches one of the two eigenvalues directly. Options (A) 1, (B) 3, and (D) 7 do not satisfy \(\lambda^2=25\), so none of them solve the characteristic equation.

Step 4: Final Answer.
One of the eigenvalues is 5, which is option (C). \[ \boxed{5} \]