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Counting matrices with entries from a set satisfying a condition on the entries

This type of question, asked 7 times: 2021 Paper 1, 2021 Paper 5, 25 Jun 2022, 27 Jun 2022, 29 Jul 2022, 31 Jan 2023, 24 Jan 2026 S1. All 7 questions below, oldest first.

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  1. Question 1

    2021 Paper 1

    90
    Let MM be any 3×33 \times 3 matrix with entries from the set {0,1,2}\mathbf{\{ 0,1,2\}}. The maximum number of such matrices, for which the sum of diagonal elements of M⊤MM^{\top}M is seven, is

  2. Question 2

    2021 Paper 5

    85
    The total number of 3×33 \times 3 matrices AA having entries from the set {0,1,2,3}\{0,1,2,3\} such that the sum of all the diagonal entries of AATAA^{T} is 9, is equal to

  3. Question 3

    25 Jun 2022

    89
    Let AA be a 3×33 \times 3 matrix having entries from. the set {−1,0,1}\mathbf{\{ - 1,0,1\}}. The number of all such matrices A having sum of all the entries equal to 5, is

  4. Question 4

    27 Jun 2022

    173
    Let A be a matrix of order 2×22 \times 2, whose entries are from the set {0,1,2,3,4,5}\mathbf{\{ 0,1,2,3,4,5\}}. If the sum of all the entries of AA is a prime number p,2<p<8p,2 <p< 8, then the number of such matrices AA is:

  5. Question 5

    29 Jul 2022

    86
    The number of matrices of order 3×33 \times 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is

  6. Question 6

    31 Jan 2023

    171
    Let A=[aij],aij∈Z∩[0,4],1≤i,j≤2A =\left\lbrack a_{ij} \right\rbrack,a_{ij}\in Z \cap \lbrack 0,4\rbrack,1 \leq i,j \leq 2. The number of matrices AA such that the sum of all entries is a prime number p∈(2,13)p \in (2,13) is

  7. Question 7

    24 Jan 2026 S1

    75
    The number of 3×23 \times 2 matrices A , which can be formed using the elements of the set {−2,−1,0,1,2}\{ - 2, - 1,0,1,2\} such that the sum of all the diagonal elements of ATAA^{T}A is 5, is ____\_\_\_\_