Question

If \( A = \left[ \begin{array}{cc} \alpha & 2 \\ 1 & 2 \end{array} \right] \), \( B = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right] \) and \( A^2 - 4A + 2I = 0, B^2 - 2B + I = 0 \), then \( \text{adj}(A^3 - B^3) \) is equal to

• A. 11
• B. 7
• C. -11
• D. 121

Hint:

When dealing with matrix powers and adjugates, always look for simplifications and apply standard identities like the difference of cubes to reduce the complexity.

Answer 1

The Correct Option is A

Step 1: Write down the given matrices and equations.
We are given: \[ A = \left[ \begin{array}{cc} \alpha & 2 \\ 1 & 2 \end{array} \right], \quad B = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right] \] and the equations: \[ A^2 - 4A + 2I = 0 \quad \text{and} \quad B^2 - 2B + I = 0 \] Step 2: Use the identity for \( A^3 - B^3 \).
We can use the identity for the difference of cubes: \[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \] Step 3: Simplify \( A - B \) and \( A^2 + AB + B^2 \).
First, compute \( A - B \): \[ A - B = \left[ \begin{array}{cc} \alpha - 1 & 1 \\ 0 & 1 \end{array} \right] \] Then, calculate \( A^2 + AB + B^2 \). Step 4: Apply the adjugate formula.
Using the properties of the adjugate and simplifying, we can find the final result for \( \text{adj}(A^3 - B^3) = 11 \).