(HP71B) integer determinant

[Albert Chan]

OP last example, step by step. (pivot taken from top left corner)
Because of delayed divison, all intermediate matrix cells are integers.

\(
\begin{vmatrix}
13 & 72 & 57 & 94 & 90 & 92 & 35 \\
40 & 93 & 90 & 99 & 1 & 95 & 66 \\
48 & 91 & 71 & 48 & 93 & 32 & 67 \\
7 & 93 & 29 & 2 & 24 & 24 & 7 \\
41 & 84 & 44 & 40 & 82 & 27 & 49 \\
3 & 72 & 6 & 33 & 97 & 34 & 4 \\
43 & 82 & 66 & 43 & 83 & 29 & 61
\end{vmatrix} \)

\( =
\begin{vmatrix}
-1671 & -1110 & -2473 & -3587 & -2445 & -542 \\
-2273 & -1813 & -3888 & -3111 & -4000 & -809 \\
705 & -22 & -632 & -318 & -332 & -154 \\
-1860 & -1765 & -3334 & -2624 & -3421 & -798 \\
720 & -93 & 147 & 991 & 166 & -53 \\
-2030 & -1593 & -3483 & -2791 & -3579 & -712
\end{vmatrix} ÷ (13)^5 \)

\( =
\begin{vmatrix}
38961 & 67363 & -227290 & 86655 & 9221 \\
63024 & 215349 & 235401 & 175269 & 49188 \\
68055 & 74718 & -175932 & 89907 & 25026 \\
73431 & 118071 & 71283 & 114078 & 36831 \\
31431 & 61531 & -201373 & 78243 & 6884
\end{vmatrix} ÷ (-1671)^4 \)

\( =
\begin{vmatrix}
-2480387 & -14061151 & -818259 & -799084 \\
1001377 & -5154838 & 1432938 & -207961 \\
207282 & -11650143 & 1148157 & -453540 \\
-167578 & 419953 & -194358 & 12937
\end{vmatrix} ÷ (38961)^3 \)

\( =
\begin{vmatrix}
689574353 & -70194683 & 33777575 \\
816495643 & -68742161 & 33125188 \\
-87215049 & 8854004 & -4260611
\end{vmatrix} ÷ (-2480387)^2 \)

\( =
\begin{vmatrix}
-3995675528 & 1909767603 \\
6667765 & -3186916
\end{vmatrix} ÷ (689574353)^1 \)

\( =
\begin{vmatrix}
1 \\
\end{vmatrix} ÷ (-3995675528)^0 = 1 \)

Ref: Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination, by Erwin H. Bareiss