Question

For some \(\alpha, \beta \in R\), let \(A = \begin{pmatrix} \alpha & 2 \\1 & 2 \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 1 \\ \beta & 1 \end{pmatrix}\) be such that \(A^2-4A+2I-B^2-3B+I=O\). Then \((\det(\text{adj}(A^3-B^3)))^2\) is equal to:

Hint:

When faced with a question that seems to have a typo or contradiction, it's important to show why it's flawed.
In an exam where a specific numerical answer is expected, try to deduce the simplest possible intended question that would lead to such an answer.
Working backwards from the answer is a valid strategy in these rare cases.

Answer 1

Correct Answer: 50625

Step 1: Understanding the Question:
We are given a matrix equation involving two \(2 \times 2\) matrices \(A\) and \(B\), defined in terms of parameters \(\alpha\) and \(\beta\).
The objectives are:

• Determine the values of \(\alpha\) and \(\beta\),
• Compute the matrix \(M = A^3 - B^3\),
• Evaluate \((\det(\operatorname{adj}(M)))^2\).

Step 2: Key Results and Properties:

• Use the matrix equation \[ A^2 - 4A + 3I = B^2 + 3B \] and equate corresponding elements.
• For an \(n \times n\) matrix \(M\), \[ \det(\operatorname{adj}(M)) = (\det M)^{\,n-1}. \] Since \(M\) is \(2 \times 2\), \[ \det(\operatorname{adj}(M)) = \det(M). \]
• Hence, the required quantity reduces to \[ (\det(A^3 - B^3))^2. \]

Step 3: Detailed Analysis:
Evaluating the Given Matrix Equation
The equation is: \[ A^2 - 4A + 3I = B^2 + 3B. \] Computing the left-hand side: \[ A^2 = \begin{pmatrix} \alpha^2 + 2 & 2\alpha + 4 \\ \alpha + 2 & 6 \end{pmatrix}. \] Thus, \[ A^2 - 4A + 3I = \begin{pmatrix} \alpha^2 - 4\alpha + 5 & 2\alpha - 4 \\ \alpha - 2 & 1 \end{pmatrix}. \] Computing the right-hand side: \[ B^2 = \begin{pmatrix} 1 + \beta & 2 \\ 2\beta & \beta + 1 \end{pmatrix}, \quad B^2 + 3B = \begin{pmatrix} \beta + 4 & 5 \\ 5\beta & \beta + 4 \end{pmatrix}. \] Equating corresponding elements: \[ 1 = \beta + 4 \quad \Rightarrow \quad \beta = -3, \] \[ 2\alpha - 4 = 5 \quad \Rightarrow \quad \alpha = \frac{9}{2}. \] Checking consistency using the \((2,1)\) entry: \[ \alpha - 2 = \frac{5}{2}, \qquad 5\beta = -15. \] Since \(\frac{5}{2} \ne -15\), the system is inconsistent. Therefore, the given question contains a **mathematical contradiction**.
Assumed Intended Interpretation
Since the published answer is \(50625\), we infer that the intended result satisfies: \[ (\det(A^3 - B^3))^2 = 50625. \] This implies: \[ \det(A^3 - B^3) = \pm 225. \] A natural scenario is: \[ A^3 - B^3 = 15I. \] Then: \[ M = A^3 - B^3 = \begin{pmatrix} 15 & 0 \\ 0 & 15 \end{pmatrix}. \] Hence: \[ \det(M) = 15^2 = 225. \] Since \(M\) is \(2 \times 2\), \[ \det(\operatorname{adj}(M)) = \det(M) = 225. \] Therefore: \[ (\det(\operatorname{adj}(M)))^2 = 225^2 = 50625. \]
Step 4: Final Answer:
The given problem statement is internally inconsistent. However, assuming the intended result was \(A^3 - B^3 = 15I\), the required value is: \[ \boxed{50625}. \]