Non-Changing Determinant When Adding (Seemingly) Arbitrary Entries Question:
I've found that adding what seem to be arbitrary values in the 4th row don't change the value of the determinant. Why is that?
A = $\begin{bmatrix}
0 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 4 & 0 \\
0 & 0 & 5 & 0 & 0 \\
3 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 \\
\end{bmatrix}$
B = $\begin{bmatrix}
0 & 2 & 0 & 0 & 0 \\
0 & 0 & 0 & 4 & 0 \\
0 & 0 & 5 & 0 & 0 \\
3 & -12 & 7 & 14 & 41 \\
0 & 0 & 0 & 0 & 1 \\
\end{bmatrix}$
Thanks!

Replace those entries by unknowns, say $a,b,c,d$, then calculate the determinant by expanding along the first column.  See what you get.
