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Find unknown entries from a quadratic matrix equation, then evaluate

This type of question, asked 2 times: 21 Jan 2026 S1, 2 Apr 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    21 Jan 2026 S1

    74
    For some α,β∈R\alpha,\beta \in R, let A=[α212]A = \begin{bmatrix} \alpha & 2 \\ 1 & 2 \end{bmatrix} and B=[111β]B = \begin{bmatrix} 1 & 1 \\ 1 & \beta \end{bmatrix} be such that A2−4A+2I=B2−3B+I=OA^{2} - 4A + 2I = B^{2} - 3B + I = O. Then (det⁡(adj(A3−B3)))2\left( \det\left( adj\left( A^{3} - B^{3} \right) \right) \right)^{2} is equal to ____\_\_\_\_

  2. Question 2

    2 Apr 2026 S1

    54
    Let A=[121α]A=\begin{bmatrix} 1 & 2 \\ 1 & \alpha \end{bmatrix} and B=[33β2]B=\begin{bmatrix} 3 & 3 \\ \beta & 2 \end{bmatrix}. If A2−4A+I=0A^{2}- 4A+I= 0 and B2−5B−6I=0B^{2}- 5B- 6I= 0, then among the two statements : (S1) : [(B−A)(B+A)]T=[1315710]\lbrack(B - A)(B + A)\rbrack^{T}=\begin{bmatrix} 13 & 15 \\ 7 & 10 \end{bmatrix} and (S2) : det(adj(A+B))=−5det(adj(A + B)) = - 5,

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