Question

If \(A = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix}\) and \( A^2 - kA - 5I = 0 \), then the value of \( k \) is:

• A. \( 3 \)
• B. \( 5 \)
• C. \( 7 \)
• D. \( 9 \)

Hint:

For matrix equations, always compute \( A^2 \) explicitly and use the given identity equation to find unknown parameters.

Answer 1

The Correct Option is B

Step 1: Characteristic Equation. We substitute \( A \) into the given equation: \[ A^2 - kA - 5I = 0 \] Calculating \( A^2 \): 

\[A^2 = \begin{bmatrix} 1 & 3\\  3 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 10 & 15 \\ 15 & 25 \end{bmatrix}\]

 Using matrix identity \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\), substituting in the equation and solving for \( k \), we obtain: \[ k = 5 \] 

Conclusion: Thus, the required value is \( 5 \), which corresponds to option \( \mathbf{(B)} \).