Generate a $5 × 5$ matrix such that the each entry is an integer between $1$ and $9$, inclusive, and whose determinant is divisible by $271$.
This is a practice problem for a linear algebra exam I have coming up, and can't for the life of me figure it out. I was thinking maybe making a triangular matrix ($0$s below main diagonal, determinant would be the product of the diagonal), but that wouldn't work because it says to use integers $1$-$9$.
Been thinking of this problem all day. The only way we've covered the determinant for a large matrix ($3 × 3$ or larger) has been through summing up the signed elementary product, but for a $5 × 5$ matrix, you'd need to make sure that all $5! = 120$ signed elementary products would need to be divisible by $271$.
If anyone has a better way to approach and solve this problem, it would be very much appreciated.