# Math Help - Prove that the det of a Matrix is divisible by 19 given...

1. ## Prove that the det of a Matrix is divisible by 19 given...

So, I'm given this question that I am REALLY stuck on, I honestly have no idea where to begin, it's got me absolutely stumped and none of the information I have on hand helps me out (Unless I'm overlooking something simple). Anyway, the question is:

The five numbers 21,375 38,798 34,162 40,223 and 79,154 are all divisible by 19. Prove that the determinant of the matrix:

$\begin{pmatrix}2&1&3&7&5\\3&8&7&9&8\\3&4&1&6&2\\4& 0&2&2&3\\7&9&1&5&4\\\end{pmatrix}$

is also divisible by 19

Any help would be appreciated

2. Originally Posted by AlmightyMidget
So, I'm given this question that I am REALLY stuck on, I honestly have no idea where to begin, it's got me absolutely stumped and none of the information I have on hand helps me out (Unless I'm overlooking something simple). Anyway, the question is:

The five numbers 21,375 38,798 34,162 40,223 and 79,154 are all divisible by 19. Prove that the determinant of the matrix:

$\begin{pmatrix}2&1&3&7&5\\3&8&7&9&8\\3&4&1&6&2\\4& 0&2&2&3\\7&9&1&5&4\\\end{pmatrix}$

is also divisible by 19

Any help would be appreciated
do this column operation and you'll be done (here $C_j$ is the j-th column): $10000C_1 + 1000C_2 + 100C_3 + 10C_4+C_5 \to C_5.$

3. How did tou figure it out? Is there a general way to do it

4. Originally Posted by pankaj

How did tou figure it out? Is there a general way to do it
i think if you do what i suggested in my previous post, you'll see how i figured it out.

5. Thanks very much NonCommAlg! I did what you suggested during my break today, and I almost kicked myself, it seems so simple, thanks!

6. Originally Posted by AlmightyMidget
So, I'm given this question that I am REALLY stuck on, I honestly have no idea where to begin, it's got me absolutely stumped and none of the information I have on hand helps me out (Unless I'm overlooking something simple). Anyway, the question is:

The five numbers 21,375 38,798 34,162 40,223 and 79,154 are all divisible by 19. Prove that the determinant of the matrix:

$\begin{pmatrix}2&1&3&7&5\\3&8&7&9&8\\3&4&1&6&2\\4& 0&2&2&3\\7&9&1&5&4\\\end{pmatrix}$

is also divisible by 19

Any help would be appreciated
Wait the 5 numbers are ordered pairs: 21,375 38,798 34,162 40,223 and 79,154?