Question

If \( A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \) satisfies the matrix polynomial equation \( A^2 - 4 + kI_2 = 0 \), then determine the value of \( k \).

• A. \( 2 \)
• B. \( 1 \)
• C. \( 3 \)
• D. \( 0 \)

Hint:

For matrix polynomial equations, first calculate \( A^2 \) and then use the properties of the identity matrix to solve for the unknowns.

Answer 1

The Correct Option is B

Step 1: Formulate the equation using the provided matrix.
Given the equation \( A^2 - 4 + kI_2 = 0 \), we first compute \( A^2 \) for the matrix \( A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \). The calculation of \( A^2 \) is as follows: \[ A^2 = A \times A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & -4 \\ -4 & 5 \end{bmatrix} \] Step 2: Substitute into the equation.
\[ A^2 - 4I_2 + kI_2 = \begin{bmatrix} 5 & -4 \\ -4 & 5 \end{bmatrix} - 4I_2 + kI_2 \] \[ = \begin{bmatrix} 5 & -4 \\ -4 & 5 \end{bmatrix} - \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix} + \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix} \] \[ = \begin{bmatrix} 1 + k & -4 \\ -4 & 1 + k \end{bmatrix} \] Step 3: Solve for \(k\).
For the matrix to equal zero, we require: \[ 1 + k = 0 \quad \text{and} \quad -4 = 0 \] However, the second condition, \(-4 = 0\), is not possible.

Conclusion: Therefore, the equation \(A^2 - 4 + kI_2 = 0\) has no real solution for \(k\).