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Determinant of P inverse A P - kI (similar matrices have the same determinant)

This type of question, asked 2 times: 2021 Paper 5, 29 Jan 2024. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 5

    86
    Let
    P=[−302056901401121206014] and A=[27ω2−1−ω10−ω−ω+1]P=\begin{bmatrix} - 30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{bmatrix}\text{~and~}A=\begin{bmatrix} 2 & 7 & \omega^{2} \\ - 1 & -\omega & 1 \\ 0 & -\omega & -\omega+ 1 \end{bmatrix}
    where ω=−1+i32\omega=\frac{- 1 +i\sqrt{3}}{2}, and I3I_{3} be the identity matrix of order 3. If the determinant of the matrix (P−1AP−I3)2\left( P^{- 1}AP -I_{3} \right)^{2} is αω2\alpha\omega^{2}, then the value of α\alpha is equal to

  2. Question 2

    29 Jan 2024

    151
    Let A=[2126211332]A =\begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix} and P=[120502715]P =\begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}. The sum of the prime factors of ∣P−1AP−2I∣\left| P^{- 1}AP - 2I \right| is equal to

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