[Math] determinant divisible by 13

linear algebramatrices

Question:
Given: $195,403$ and $247$ are divisible by 13.

Prove (without actually calculating the determinant) that

$$\det \begin{bmatrix} 1 & 9 & 5 \\ 2 & 4 & 7 \\ 4 & 0 & 3 \end{bmatrix}$$

is divisible by 13.

What I did:

Apart from calculating the determinant and seeing that it's true, I couldn't really think of anything else…

Thanks

Best Answer

Hint: If you multiply the first column by $100$, how does that change the determinant? Then if you add the third column to the first, how does that change the determinant?

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