how do I prove that the matrix certain numbers has a determinant that is divisible by the same common divisor of said numbers?

Question

Recently, I have been given the following task, which is practically unsolvable for me: The numbers 2014, 1484, 3710 and 6996 are all divisible by 106. Without calculating it, show that the determinant of \begin{pmatrix}2&0&1&4\\ \:\:1&4&8&4\\ \:\:3&7&1&0\\ \:\:6&9&9&6\end{pmatrix} is also divisible by 106. I simply don't know where to begin, I have tried reducing the matrix, finding a similar matrix, finding some 2 matrices that are divisible by 106 and if multiplied with each other yield the given matrix, but everything as failed so far. Can anyone give me a hint, what I could use to make this prove?

Answer 1

If your matrix is $M$, note that $$ M \pmatrix{1000\cr 100\cr 10\cr 1} = \pmatrix{2014\cr 1484\cr 3710\cr 6996}$$ The fact that each term on the right is divisible by $106$ tells you that your matrix is singular modulo certain primes...

Answer 2

Splicing the answer from here. Number the columns $C_1,C_2,C_3,C_4$ from left to right. The operation $$C_4\leftarrow 1000C_1+100C_2+10C_3+C_4$$ does not change the determinant, but leaves the last column having numbers all divisible by $106$ (per the givens in the question). If now we divide the last column by $106$, the determinant is divided by the same amount, and we still have an integer matrix with integer determinant. Thus the original matrix has a determinant divisible by $106$.

Answer 3

The matrix $$ \pmatrix{2&0&1&4\\1&4&8&4\\3&7&1&0\\6&9&9&6} \pmatrix{1&0&0&1000\\0&1&0&100\\0&0&1&10\\0&0&0&1} = \pmatrix{2&0&1&2014\\1&4&8&1484\\3&7&1&3710\\6&9&9&6996} $$ has the same determinant. Dividing the determinant of this matrix by $106$ gives us the determinant of the matrix $$ \pmatrix{2&0&1&2014/106\\1&4&8&1484/106\\3&7&1&3710/106\\6&9&9&6996/106}. $$ This is a matrix with integer entries, so this determinant is an integer.

Equivalently: in terms of column operations, the operations $$ C_4 \to 1000 C_1 + 100 C_2 + 10 C_3 + C_4\\ C_4 \to \frac 1{106}C_4 $$ gives us a matrix whose determinant is an integer $k$. It follows that the determinant of the original matrix is equal to $106k$.