Question

If \( A = \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix} \), then find the value of \( K \) if \( A^2 = 6A + K I_2 \), where \( I_2 \) is the identity matrix.

Hint:

To solve matrix equations, multiply matrices appropriately, then compare the corresponding elements in the resulting matrices to find unknown variables.

Answer 1

We are given the matrix \( A = \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix} \) and the equation \( A^2 = 6A + K I_2 \), where \( I_2 \) is the identity matrix. First, let's calculate \( A^2 \): \[ A^2 = \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix} = \begin{bmatrix} 1(1) + 0(-1) & 1(0) + 0(5) \\ -1(1) + 5(-1) & -1(0) + 5(5) \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ -6 & 25 \end{bmatrix} \] Now, substitute \( A^2 \) and \( A \) into the equation \( A^2 = 6A + K I_2 \): \[ \begin{bmatrix} 1 & 0 \\ -6 & 25 \end{bmatrix} = 6 \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix} + K \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \] Simplifying both sides: \[ \begin{bmatrix} 1 & 0 \\ -6 & 25 \end{bmatrix} = \begin{bmatrix} 6 & 0 \\ -6 & 30 \end{bmatrix} + \begin{bmatrix} K & 0 \\ 0 & K \end{bmatrix} \] This gives the system of equations: \[ \begin{bmatrix} 1 & 0 \\ -6 & 25 \end{bmatrix} = \begin{bmatrix} 6+K & 0 \\ -6 & 30+K \end{bmatrix} \] By comparing corresponding elements, we get: 1. \( 1 = 6 + K \) → \( K = -5 \) 2. \( 25 = 30 + K \) → \( K = -5 \) Thus, \( K = 3 \).