This is a relatively sparse matrix, so we might be able to solve it directly.

1. Find any row or column that has only one non-zero value. The remaining zeros nullify any sub-matrix they might multiply, so the determinant must go through that value. Accordingly, we can zero out the rest of the column or row (note the reversal of terms) where a value is the only non-zero in its column or row. (There are two such.)
2. Iterate #1 as far as possible. (This isolates a third row & column)
3. The determinant of the remaining matrix is (relatively) easy to compute.