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Number of matrices with entries from a set whose sum of all entries is a prime

The same question, asked 3 times: 27 Jun 2022*, 29 Jul 2022*, 31 Jan 2023. Every wording below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    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:

  2. Question 2

    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

  3. Question 3

    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