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Count matrices with entries from a set when the trace of A^T A or A^2 is given

This type of question, asked 2 times: 2021 Paper 25, 27 Jul 2022. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 25

    71
    Let AA be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A2A^{2} is 1 , then the possible number of such matrices is:

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  2. Question 2

    27 Jul 2022

    88
    Let SS be the set containing all 3×33 \times 3 matrices with entries from {−1,0,1}\mathbf{\{ - 1,0,1\}}. The total number of matrices A∈SA \in S such that the sum of all the diagonal elements of ATAA^{T}A is 6 is