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A is non-singular and (A - 3I)(A - 5I) = O: express A^-1 in terms of A

The same question, asked 2 times: 4 May 2023 S1, 26 Apr 2025 S1. Every wording below, oldest first.

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  1. Question 1

    4 May 2023 S1

    101
    Suppose A is any 3×33\times3 non-singular matrix and (A−3I)(A−5I)=O(A-3I)(A-5I)=O where I=I3I=I_3 and O=O3O=O_3. Here O3O_3 represent zero matrix of order 3 and I3I_3 is an identity matrix of order 3. If αA+βA−1=4I\alpha A + \beta A^{-1} = 4I, then α+β\alpha + \beta is equal to

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  2. Question 2

    26 Apr 2025 S1

    143
    Matrix A is non-singular matrix and (A−3I)(A−5I)=0(A- 3I)(A- 5I) = 0, then 158A−1=\frac{15}{8}A^{- 1}= ____\_\_\_\_

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