TITLE: Prove that the determinant is a multiple of $17$ without developing it
QUESTION [6 upvotes]: Let, matrix is given as :
$$D=\begin{bmatrix}
1 & 1 & 9 \\
1 & 8 & 7 \\
1 & 5 & 3\end{bmatrix}$$

Prove that the determinant is a multiple of $17$ without developing it?


I saw a resolution by the Jacobi method , but could not apply the methodology in this example.

REPLY [3 votes]: Notice that 119, 187 and 153 are all divisible by 17. So multiplying column 2 by 10 and adding to column 3 and multiplying column 1 by 100 and adding to column 3, gives us a column in which each element is divisible by 17:
$D=\left|\begin{matrix}
1 & 1 & 9 \\
1 & 8 & 7 \\
1 & 5 & 3\end{matrix}\right|
=\left|\begin{matrix}
1 & 1 & 19 \\
1 & 8 & 87 \\
1 & 5 & 53\end{matrix}\right|
=\left|\begin{matrix}
1 & 1 & 119 \\
1 & 8 & 187 \\
1 & 5 & 153\end{matrix}\right|
=17\left|\begin{matrix}
1 & 1 & 7 \\
1 & 8 & 11 \\
1 & 5 & 9\end{matrix}\right|$
Thus $D = 17\cdot E$ where $E$ is the determinant of a matrix whose elements are integers which multiplied out using the definition of a determinant will be an integer.