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: Write the equation using the given matrix.
We are given the equation \( A^2 - 4 + kI_2 = 0 \). Let’s first calculate \( A^2 \) for the matrix \( A \). \[ A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \] Now, calculate \( A^2 \): \[ A^2 = A \times A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & -4 \\ -4 & 5 \end{bmatrix} \] Step 2: Substituting in the equation. 
\[ A^2 - 4 + kI_2 = \begin{bmatrix} 5 & -4 \\ -4 & 5 \end{bmatrix} - 4I_2 + kI_2 \] \[ = \begin{bmatrix} 5 & -4 \\ -4 & 5 \end{bmatrix} - \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix} + \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix} \] \[ = \begin{bmatrix} 1 + k & -4 \\ -4 & 1 + k \end{bmatrix} \] Step 3: Solve for \(k\). 
For the matrix to be zero: \[ 1 + k = 0 \quad \text{and} \quad -4 = 0 \] The second condition is impossible (\(-4 \neq 0\)). 

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