If \(A = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix}\) and \( A^2 - kA - 5I = 0 \), then the value of \( k \) is:
Show Hint
- \( 3 \)
- \( 5 \)
- \( 7 \)
- \( 9 \)
The Correct Option is B
Solution and Explanation
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 \( \mathbf{(B)} \).
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