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Reduce a higher power of A using a given polynomial relation (such as A^2 = 3A + alpha I)

This type of question, asked 5 times: 2021 Paper 10, 29 Jan 2023, 6 Apr 2023, 2 Apr 2025, 4 Apr 2025. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 10

    82
    Let I be an identity matrix of order 2×22 \times 2 and P=[2−15−3]P =\begin{bmatrix} 2 & - 1 \\ 5 & - 3 \end{bmatrix}. Then the value of n∈Nn \in N for which Pn=5I−8PP^{n}= 5I - 8P is equal to

  2. Question 2

    29 Jan 2023

    61
    Let α\alpha and β\beta be real numbers. Consider a 3×33 \times 3 matrix A such that A2=3A+αIA^{2}= 3A + \alpha I. If A4=21A+βIA^{4}= 21A + \beta I, then

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  3. Question 3

    6 Apr 2023

    160
    Let PP be a square matrix such that P2=I−PP^{2}= I - P. For α,β,γ,δ∈N\alpha,\beta,\gamma,\delta \in N, if Pα+Pβ=γI−29PP^{\alpha}+P^{\beta}= \gamma I - 29P and Pα−Pβ=P^{\alpha}-P^{\beta}= δI−13P\delta I - 13P, then α+β+γ−δ\alpha + \beta + \gamma - \delta is equal to

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  4. Question 4

    2 Apr 2025

    145
    Let AA be a 3×33 \times 3 real matrix such that A2(A−2I)−A^{2}(A- 2I) - 4( A−I)=O4(\text{ }A - I) = O, where I and O are the identity and null matrices, respectively. If A5=αA2+βA+γIA^{5}=\alpha A^{2}+\beta A+\gamma I, where α,β\alpha,\beta and γ\gamma are real constants, then α+β+γ\alpha+\beta+\gamma is equal to:

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  5. Question 5

    4 Apr 2025

    136
    Let the matrix A=[100101010]A=\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} satisfy An=An−2+A2−A^{n}=A^{n- 2}+A^{2}- I for n≥3n \geq 3. Then the sum of all the elements of A50A^{50} is :-

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