If A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} and A^2 - kA - I_2 = 0, then value of k is-
Step-by-Step Solution
Key Concept: Calculate A^2, substitute into the given equation, and solve for k by comparing corresponding elements.
Given A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}. A^2 = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 1+4 & 2+6 \\ 2+6 & 4+9 \end{bmatrix} = \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix}. The equation is A^2 - kA - I_2 = 0, so \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix} - k \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}. This gives 5 - k - 1 = 0, so k = 4.
Correct Answer: 4