Is there any easy way to calculate the value of this determinant?

Question

\begin{vmatrix} 0 & 3 & 1 & 2 & 10! & e^{-7}\\ 1 & 2 & -1 & 2 & \sqrt{2} & 2 \\ -1 & -2 & 3 & -3 & 1 & -\frac{1}{5} \\ -2 & -1 & 3 & 2 & -2 & -9 \\ 0 & 0 & 0 & 0 & 4 & 2 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ \end{vmatrix}

I still can't see any easy way to compute the determinant above I would appreciate any kind of help. Thanks.

Answer 1

Remember that you can add a multiple of a row / column to another row / column, and it does not change the determinant. Then use the fact that some rows are similar. For instance, do the following operations in order:

• $R_5 \leftarrow R_5 - 4 R_6$
• $R_3 \leftarrow R_3 + R_2$
• $R_4 \leftarrow R_4 + 2 R_2$
• $R_4 \leftarrow R_4 - R_1$

Then flip $R_6$ and $R_5$, $R_1$ and $R_2$ (each operation multiplies the determinant by $-1$, so it remains unchanged), and you get a triangular matrix with diagonal $(1, 3, 2, 4, 1, -2)$, so you have a determinant of $- 48$.