<s>[INST] When does this "primes" matrix have full row and column rank? When does the following type of matrix have full row and column rank?
For each entry, we take the value $v$=row*column.
Then for the $b$th bit of $v$, $v_b$ we take the $b$th prime to the $v_b$ power.
EXAMPLES
Say row = 5 and column = 7.
This is 5*7=35, which, in binary, is $100011_2$
So we take
$$\cdot 13^1 \cdot 11^0 \cdot 7^0 \cdot 5^0 \cdot 3^1 \cdot 2^1$$ 
$$=2 \cdot 3 \cdot 13$$
$$=78$$
...So the entry for the 5th row and 7th column would be 78.
Say row = 4 and column = 2.
This is 4*2=8, which, in binary, is $001000_2$
So we take
$$\cdot 13^0 \cdot 11^0 \cdot 7^1 \cdot 5^0 \cdot 3^0 \cdot 2^0$$ 
$$=7$$
...So the entry for the 4th row and 2nd column would be 7.
THE GENERAL MATRIX
$$
        \begin{matrix}
        2^0 3^0 5^0  &  2^0 3^0 5^0  &  2^0 3^0 5^0  &  2^0 3^0 5^0  &  2^0 3^0 5^0  &  \dots\\
        2^0 3^0 5^0  &  2^1 3^0 5^0  &  2^0 3^1 5^0  &  2^1 3^1 5^0  &  2^0 3^0 5^1  &  \dots\\
        2^0 3^0 5^0  &  2^0 3^1 5^0  &  2^0 3^0 5^1  &  2^0 3^1 5^1  &  2^0 3^0 5^0 7^1  &  \dots\\
        2^0 3^0 5^0  &  2^1 3^1 5^0  &  2^1 3^1 5^0  &  2^1 3^0 5^0 7^1  &  2^0 3^0 5^1 7^1  &  \dots\\
        \vdots  &  \vdots  &  \vdots  &  \vdots  &  \vdots  &  \ddots\\
 \\
        \end{matrix}
$$
We can assume that the matrix is an $N \times N$ matrix.
SOME OBSERVATIONS
Just a general observation:  The only counterexample for $N \times N$ matrices with $N$ ranging from 2 to 20 is $N=3$.  All other matrices have full rank.
 [/INST] Easy counterexample, take $N = 3$ and you will see that it is not full ranked: 
$\left( \begin{array}{ccc}
2^03^05^0=1 & 2^03^05^0=1 & 2^03^05^0=1 \\
2^03^05^0=1 & 2^13^05^0=2 & 2^03^15^0=3 \\
2^03^05^0=1 & 2^03^15^0=3 & 2^03^05^1=5 \end{array} \right)$, this matrix is not full rank, so your question's answer would be no, it is not always full rank.
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